Finite Math Examples

$[\begin{array}{c} - 6 & 8 & 2 & 4 & - 3 \\ 1 & 9 & 7 & - 12 & 5 \\ 15 & 2 & - 8 & 10 & 11 \\ 4 & 7 & 9 & 6 & - 2 \\ 1 & 3 & 8 & 23 & 4 \end{array}] [\begin{array}{c} 5 & - 3 & 7 & 9 & 2 \\ 6 & 8 & - 5 & 2 & 1 \\ 3 & 7 & - 4 & 2 & 11 \\ 5 & - 3 & 9 & 4 & - 1 \\ 0 & 3 & 2 & 5 & 1 \end{array}]$

Step 1

Multiply $[\begin{array}{c} - 6 & 8 & 2 & 4 & - 3 \\ 1 & 9 & 7 & - 12 & 5 \\ 15 & 2 & - 8 & 10 & 11 \\ 4 & 7 & 9 & 6 & - 2 \\ 1 & 3 & 8 & 23 & 4 \end{array}] [\begin{array}{c} 5 & - 3 & 7 & 9 & 2 \\ 6 & 8 & - 5 & 2 & 1 \\ 3 & 7 & - 4 & 2 & 11 \\ 5 & - 3 & 9 & 4 & - 1 \\ 0 & 3 & 2 & 5 & 1 \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $5 \times 5$ and the second matrix is $5 \times 5$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} - 6 \cdot 5 + 8 \cdot 6 + 2 \cdot 3 + 4 \cdot 5 - 3 \cdot 0 & - 6 \cdot - 3 + 8 \cdot 8 + 2 \cdot 7 + 4 \cdot - 3 - 3 \cdot 3 & - 6 \cdot 7 + 8 \cdot - 5 + 2 \cdot - 4 + 4 \cdot 9 - 3 \cdot 2 & - 6 \cdot 9 + 8 \cdot 2 + 2 \cdot 2 + 4 \cdot 4 - 3 \cdot 5 & - 6 \cdot 2 + 8 \cdot 1 + 2 \cdot 11 + 4 \cdot - 1 - 3 \cdot 1 \\ 1 \cdot 5 + 9 \cdot 6 + 7 \cdot 3 - 12 \cdot 5 + 5 \cdot 0 & 1 \cdot - 3 + 9 \cdot 8 + 7 \cdot 7 - 12 \cdot - 3 + 5 \cdot 3 & 1 \cdot 7 + 9 \cdot - 5 + 7 \cdot - 4 - 12 \cdot 9 + 5 \cdot 2 & 1 \cdot 9 + 9 \cdot 2 + 7 \cdot 2 - 12 \cdot 4 + 5 \cdot 5 & 1 \cdot 2 + 9 \cdot 1 + 7 \cdot 11 - 12 \cdot - 1 + 5 \cdot 1 \\ 15 \cdot 5 + 2 \cdot 6 - 8 \cdot 3 + 10 \cdot 5 + 11 \cdot 0 & 15 \cdot - 3 + 2 \cdot 8 - 8 \cdot 7 + 10 \cdot - 3 + 11 \cdot 3 & 15 \cdot 7 + 2 \cdot - 5 - 8 \cdot - 4 + 10 \cdot 9 + 11 \cdot 2 & 15 \cdot 9 + 2 \cdot 2 - 8 \cdot 2 + 10 \cdot 4 + 11 \cdot 5 & 15 \cdot 2 + 2 \cdot 1 - 8 \cdot 11 + 10 \cdot - 1 + 11 \cdot 1 \\ 4 \cdot 5 + 7 \cdot 6 + 9 \cdot 3 + 6 \cdot 5 - 2 \cdot 0 & 4 \cdot - 3 + 7 \cdot 8 + 9 \cdot 7 + 6 \cdot - 3 - 2 \cdot 3 & 4 \cdot 7 + 7 \cdot - 5 + 9 \cdot - 4 + 6 \cdot 9 - 2 \cdot 2 & 4 \cdot 9 + 7 \cdot 2 + 9 \cdot 2 + 6 \cdot 4 - 2 \cdot 5 & 4 \cdot 2 + 7 \cdot 1 + 9 \cdot 11 + 6 \cdot - 1 - 2 \cdot 1 \\ 1 \cdot 5 + 3 \cdot 6 + 8 \cdot 3 + 23 \cdot 5 + 4 \cdot 0 & 1 \cdot - 3 + 3 \cdot 8 + 8 \cdot 7 + 23 \cdot - 3 + 4 \cdot 3 & 1 \cdot 7 + 3 \cdot - 5 + 8 \cdot - 4 + 23 \cdot 9 + 4 \cdot 2 & 1 \cdot 9 + 3 \cdot 2 + 8 \cdot 2 + 23 \cdot 4 + 4 \cdot 5 & 1 \cdot 2 + 3 \cdot 1 + 8 \cdot 11 + 23 \cdot - 1 + 4 \cdot 1 \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 44 & 75 & - 60 & - 33 & 11 \\ 20 & 169 & - 164 & 18 & 105 \\ 113 & - 82 & 239 & 218 & - 55 \\ 119 & 83 & 7 & 82 & 106 \\ 162 & 20 & 175 & 143 & 74 \end{array}]$

Step 2

Find the determinant.

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Step 2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - & + \\ - & + & - & + & - \\ + & - & + & - & + \\ - & + & - & + & - \\ + & - & + & - & + \end{array} |$

Step 2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 169 & - 164 & 18 & 105 \\ - 82 & 239 & 218 & - 55 \\ 83 & 7 & 82 & 106 \\ 20 & 175 & 143 & 74 \end{array} |$

Step 2.1.4

Multiply element $a_{11}$ by its cofactor.

$44 | \begin{array}{c} 169 & - 164 & 18 & 105 \\ - 82 & 239 & 218 & - 55 \\ 83 & 7 & 82 & 106 \\ 20 & 175 & 143 & 74 \end{array} |$

Step 2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} |$

Step 2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} |$

Step 2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} |$

Step 2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} |$

Step 2.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} |$

Step 2.1.10

Multiply element $a_{14}$ by its cofactor.

$33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} |$

Step 2.1.11

The minor for $a_{15}$ is the determinant with row $1$ and column $5$ deleted.

$| \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.1.12

Multiply element $a_{15}$ by its cofactor.

$11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.1.13

Add the terms together.

$44 | \begin{array}{c} 169 & - 164 & 18 & 105 \\ - 82 & 239 & 218 & - 55 \\ 83 & 7 & 82 & 106 \\ 20 & 175 & 143 & 74 \end{array} | - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2

Evaluate $| \begin{array}{c} 169 & - 164 & 18 & 105 \\ - 82 & 239 & 218 & - 55 \\ 83 & 7 & 82 & 106 \\ 20 & 175 & 143 & 74 \end{array} |$ .

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Step 2.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 2.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} |$

Step 2.2.1.4

Multiply element $a_{11}$ by its cofactor.

$169 | \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} |$

Step 2.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} |$

Step 2.2.1.6

Multiply element $a_{12}$ by its cofactor.

$164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} |$

Step 2.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} |$

Step 2.2.1.8

Multiply element $a_{13}$ by its cofactor.

$18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} |$

Step 2.2.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |$

Step 2.2.1.10

Multiply element $a_{14}$ by its cofactor.

$- 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |$

Step 2.2.1.11

Add the terms together.

$44 (169 | \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} | + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2

Evaluate $| \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} |$ .

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Step 2.2.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.2.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.2.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.2.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.2.2.1.4

Multiply element $a_{11}$ by its cofactor.

$239 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.2.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.2.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.2.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.2.2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.2.2.1.9

Add the terms together.

$44 (169 (239 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} | - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.2

Evaluate $| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$ .

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Step 2.2.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 (239 (82 \cdot 74 - 143 \cdot 106) - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.2.2

Simplify the determinant.

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Step 2.2.2.2.2.1

Simplify each term.

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Step 2.2.2.2.2.1.1

Multiply $82$ by $74$ .

$44 (169 (239 (6068 - 143 \cdot 106) - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.2.2.1.2

Multiply $- 143$ by $106$ .

$44 (169 (239 (6068 - 15158) - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.2.2.2

Subtract $15158$ from $6068$ .

$44 (169 (239 \cdot - 9090 - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.3

Evaluate $| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$ .

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Step 2.2.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 (239 \cdot - 9090 - 218 (7 \cdot 74 - 175 \cdot 106) - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.3.2

Simplify the determinant.

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Step 2.2.2.3.2.1

Simplify each term.

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Step 2.2.2.3.2.1.1

Multiply $7$ by $74$ .

$44 (169 (239 \cdot - 9090 - 218 (518 - 175 \cdot 106) - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.3.2.1.2

Multiply $- 175$ by $106$ .

$44 (169 (239 \cdot - 9090 - 218 (518 - 18550) - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.3.2.2

Subtract $18550$ from $518$ .

$44 (169 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.4

Evaluate $| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$ .

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Step 2.2.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 (7 \cdot 143 - 175 \cdot 82)) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.4.2

Simplify the determinant.

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Step 2.2.2.4.2.1

Simplify each term.

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Step 2.2.2.4.2.1.1

Multiply $7$ by $143$ .

$44 (169 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 (1001 - 175 \cdot 82)) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.4.2.1.2

Multiply $- 175$ by $82$ .

$44 (169 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 (1001 - 14350)) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.4.2.2

Subtract $14350$ from $1001$ .

$44 (169 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 \cdot - 13349) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.5

Simplify the determinant.

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Step 2.2.2.5.1

Simplify each term.

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Step 2.2.2.5.1.1

Multiply $239$ by $- 9090$ .

$44 (169 (- 2172510 - 218 \cdot - 18032 - 55 \cdot - 13349) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.5.1.2

Multiply $- 218$ by $- 18032$ .

$44 (169 (- 2172510 + 3930976 - 55 \cdot - 13349) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.5.1.3

Multiply $- 55$ by $- 13349$ .

$44 (169 (- 2172510 + 3930976 + 734195) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.5.2

Add $- 2172510$ and $3930976$ .

$44 (169 (1758466 + 734195) + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.2.5.3

Add $1758466$ and $734195$ .

$44 (169 \cdot 2492661 + 164 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3

Evaluate $| \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} |$ .

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Step 2.2.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.2.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.2.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.2.3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.2.3.1.4

Multiply element $a_{11}$ by its cofactor.

$- 82 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.2.3.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.2.3.1.6

Multiply element $a_{12}$ by its cofactor.

$- 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.2.3.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.2.3.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.2.3.1.9

Add the terms together.

$44 (169 \cdot 2492661 + 164 (- 82 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} | - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.2

Evaluate $| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$ .

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Step 2.2.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 (- 82 (82 \cdot 74 - 143 \cdot 106) - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.2.2

Simplify the determinant.

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Step 2.2.3.2.2.1

Simplify each term.

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Step 2.2.3.2.2.1.1

Multiply $82$ by $74$ .

$44 (169 \cdot 2492661 + 164 (- 82 (6068 - 143 \cdot 106) - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.2.2.1.2

Multiply $- 143$ by $106$ .

$44 (169 \cdot 2492661 + 164 (- 82 (6068 - 15158) - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.2.2.2

Subtract $15158$ from $6068$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.3

Evaluate $| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$ .

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Step 2.2.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 (83 \cdot 74 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.3.2

Simplify the determinant.

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Step 2.2.3.3.2.1

Simplify each term.

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Step 2.2.3.3.2.1.1

Multiply $83$ by $74$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 (6142 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.3.2.1.2

Multiply $- 20$ by $106$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 (6142 - 2120) - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.3.2.2

Subtract $2120$ from $6142$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.4

Evaluate $| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$ .

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Step 2.2.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 (83 \cdot 143 - 20 \cdot 82)) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.4.2

Simplify the determinant.

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Step 2.2.3.4.2.1

Simplify each term.

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Step 2.2.3.4.2.1.1

Multiply $83$ by $143$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 (11869 - 20 \cdot 82)) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.4.2.1.2

Multiply $- 20$ by $82$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 (11869 - 1640)) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.4.2.2

Subtract $1640$ from $11869$ .

$44 (169 \cdot 2492661 + 164 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 \cdot 10229) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.5

Simplify the determinant.

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Step 2.2.3.5.1

Simplify each term.

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Step 2.2.3.5.1.1

Multiply $- 82$ by $- 9090$ .

$44 (169 \cdot 2492661 + 164 (745380 - 218 \cdot 4022 - 55 \cdot 10229) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.5.1.2

Multiply $- 218$ by $4022$ .

$44 (169 \cdot 2492661 + 164 (745380 - 876796 - 55 \cdot 10229) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.5.1.3

Multiply $- 55$ by $10229$ .

$44 (169 \cdot 2492661 + 164 (745380 - 876796 - 562595) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.5.2

Subtract $876796$ from $745380$ .

$44 (169 \cdot 2492661 + 164 (- 131416 - 562595) + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.3.5.3

Subtract $562595$ from $- 131416$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4

Evaluate $| \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} |$ .

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Step 2.2.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.2.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.2.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.2.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.2.4.1.4

Multiply element $a_{11}$ by its cofactor.

$- 82 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.2.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.2.4.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.2.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.2.4.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.2.4.1.9

Add the terms together.

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.2

Evaluate $| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$ .

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Step 2.2.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 (7 \cdot 74 - 175 \cdot 106) - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.2.2

Simplify the determinant.

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Step 2.2.4.2.2.1

Simplify each term.

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Step 2.2.4.2.2.1.1

Multiply $7$ by $74$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 (518 - 175 \cdot 106) - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.2.2.1.2

Multiply $- 175$ by $106$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 (518 - 18550) - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.2.2.2

Subtract $18550$ from $518$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.3

Evaluate $| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$ .

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Step 2.2.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 (83 \cdot 74 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.3.2

Simplify the determinant.

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Step 2.2.4.3.2.1

Simplify each term.

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Step 2.2.4.3.2.1.1

Multiply $83$ by $74$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 (6142 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.3.2.1.2

Multiply $- 20$ by $106$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 (6142 - 2120) - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.3.2.2

Subtract $2120$ from $6142$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.4

Evaluate $| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$ .

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Step 2.2.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 (83 \cdot 175 - 20 \cdot 7)) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.4.2

Simplify the determinant.

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Step 2.2.4.4.2.1

Simplify each term.

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Step 2.2.4.4.2.1.1

Multiply $83$ by $175$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 (14525 - 20 \cdot 7)) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.4.2.1.2

Multiply $- 20$ by $7$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 (14525 - 140)) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.4.2.2

Subtract $140$ from $14525$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 \cdot 14385) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.5

Simplify the determinant.

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Step 2.2.4.5.1

Simplify each term.

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Step 2.2.4.5.1.1

Multiply $- 82$ by $- 18032$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (1478624 - 239 \cdot 4022 - 55 \cdot 14385) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.5.1.2

Multiply $- 239$ by $4022$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (1478624 - 961258 - 55 \cdot 14385) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.5.1.3

Multiply $- 55$ by $14385$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (1478624 - 961258 - 791175) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.5.2

Subtract $961258$ from $1478624$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 (517366 - 791175) - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.4.5.3

Subtract $791175$ from $517366$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5

Evaluate $| \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |$ .

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Step 2.2.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.2.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.2.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.2.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.2.5.1.4

Multiply element $a_{11}$ by its cofactor.

$- 82 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.2.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.2.5.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.2.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.2.5.1.8

Multiply element $a_{13}$ by its cofactor.

$218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.2.5.1.9

Add the terms together.

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} | - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.2

Evaluate $| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$ .

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Step 2.2.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 (7 \cdot 143 - 175 \cdot 82) - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.2.2

Simplify the determinant.

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Step 2.2.5.2.2.1

Simplify each term.

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Step 2.2.5.2.2.1.1

Multiply $7$ by $143$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 (1001 - 175 \cdot 82) - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.2.2.1.2

Multiply $- 175$ by $82$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 (1001 - 14350) - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.2.2.2

Subtract $14350$ from $1001$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.3

Evaluate $| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$ .

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Step 2.2.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 (83 \cdot 143 - 20 \cdot 82) + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.3.2

Simplify the determinant.

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Step 2.2.5.3.2.1

Simplify each term.

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Step 2.2.5.3.2.1.1

Multiply $83$ by $143$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 (11869 - 20 \cdot 82) + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.3.2.1.2

Multiply $- 20$ by $82$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 (11869 - 1640) + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.3.2.2

Subtract $1640$ from $11869$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.4

Evaluate $| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$ .

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Step 2.2.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 (83 \cdot 175 - 20 \cdot 7))) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.4.2

Simplify the determinant.

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Step 2.2.5.4.2.1

Simplify each term.

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Step 2.2.5.4.2.1.1

Multiply $83$ by $175$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 (14525 - 20 \cdot 7))) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.4.2.1.2

Multiply $- 20$ by $7$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 (14525 - 140))) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.4.2.2

Subtract $140$ from $14525$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 \cdot 14385)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.5

Simplify the determinant.

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Step 2.2.5.5.1

Simplify each term.

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Step 2.2.5.5.1.1

Multiply $- 82$ by $- 13349$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (1094618 - 239 \cdot 10229 + 218 \cdot 14385)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.5.1.2

Multiply $- 239$ by $10229$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (1094618 - 2444731 + 218 \cdot 14385)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.5.1.3

Multiply $218$ by $14385$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (1094618 - 2444731 + 3135930)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.5.2

Subtract $2444731$ from $1094618$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 (- 1350113 + 3135930)) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.5.5.3

Add $- 1350113$ and $3135930$ .

$44 (169 \cdot 2492661 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 \cdot 1785817) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6

Simplify the determinant.

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Step 2.2.6.1

Simplify each term.

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Step 2.2.6.1.1

Multiply $169$ by $2492661$ .

$44 (421259709 + 164 \cdot - 694011 + 18 \cdot - 273809 - 105 \cdot 1785817) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6.1.2

Multiply $164$ by $- 694011$ .

$44 (421259709 - 113817804 + 18 \cdot - 273809 - 105 \cdot 1785817) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6.1.3

Multiply $18$ by $- 273809$ .

$44 (421259709 - 113817804 - 4928562 - 105 \cdot 1785817) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6.1.4

Multiply $- 105$ by $1785817$ .

$44 (421259709 - 113817804 - 4928562 - 187510785) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6.2

Subtract $113817804$ from $421259709$ .

$44 (307441905 - 4928562 - 187510785) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6.3

Subtract $4928562$ from $307441905$ .

$44 (302513343 - 187510785) - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.2.6.4

Subtract $187510785$ from $302513343$ .

$44 \cdot 115002558 - 75 | \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} | - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3

Evaluate $| \begin{array}{c} 20 & - 164 & 18 & 105 \\ 113 & 239 & 218 & - 55 \\ 119 & 7 & 82 & 106 \\ 162 & 175 & 143 & 74 \end{array} |$ .

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Step 2.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 2.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} |$

Step 2.3.1.4

Multiply element $a_{11}$ by its cofactor.

$20 | \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} |$

Step 2.3.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} |$

Step 2.3.1.6

Multiply element $a_{12}$ by its cofactor.

$164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} |$

Step 2.3.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} |$

Step 2.3.1.8

Multiply element $a_{13}$ by its cofactor.

$18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} |$

Step 2.3.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |$

Step 2.3.1.10

Multiply element $a_{14}$ by its cofactor.

$- 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |$

Step 2.3.1.11

Add the terms together.

$44 \cdot 115002558 - 75 (20 | \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} | + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2

Evaluate $| \begin{array}{c} 239 & 218 & - 55 \\ 7 & 82 & 106 \\ 175 & 143 & 74 \end{array} |$ .

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Step 2.3.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.3.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.3.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.3.2.1.4

Multiply element $a_{11}$ by its cofactor.

$239 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.3.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.3.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.3.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.3.2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.3.2.1.9

Add the terms together.

$44 \cdot 115002558 - 75 (20 (239 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} | - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.2

Evaluate $| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$ .

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Step 2.3.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 (239 (82 \cdot 74 - 143 \cdot 106) - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.2.2

Simplify the determinant.

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Step 2.3.2.2.2.1

Simplify each term.

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Step 2.3.2.2.2.1.1

Multiply $82$ by $74$ .

$44 \cdot 115002558 - 75 (20 (239 (6068 - 143 \cdot 106) - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.2.2.1.2

Multiply $- 143$ by $106$ .

$44 \cdot 115002558 - 75 (20 (239 (6068 - 15158) - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.2.2.2

Subtract $15158$ from $6068$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.3

Evaluate $| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$ .

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Step 2.3.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 (7 \cdot 74 - 175 \cdot 106) - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.3.2

Simplify the determinant.

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Step 2.3.2.3.2.1

Simplify each term.

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Step 2.3.2.3.2.1.1

Multiply $7$ by $74$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 (518 - 175 \cdot 106) - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.3.2.1.2

Multiply $- 175$ by $106$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 (518 - 18550) - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.3.2.2

Subtract $18550$ from $518$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.4

Evaluate $| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$ .

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Step 2.3.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 (7 \cdot 143 - 175 \cdot 82)) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.4.2

Simplify the determinant.

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Step 2.3.2.4.2.1

Simplify each term.

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Step 2.3.2.4.2.1.1

Multiply $7$ by $143$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 (1001 - 175 \cdot 82)) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.4.2.1.2

Multiply $- 175$ by $82$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 (1001 - 14350)) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.4.2.2

Subtract $14350$ from $1001$ .

$44 \cdot 115002558 - 75 (20 (239 \cdot - 9090 - 218 \cdot - 18032 - 55 \cdot - 13349) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.5

Simplify the determinant.

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Step 2.3.2.5.1

Simplify each term.

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Step 2.3.2.5.1.1

Multiply $239$ by $- 9090$ .

$44 \cdot 115002558 - 75 (20 (- 2172510 - 218 \cdot - 18032 - 55 \cdot - 13349) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.5.1.2

Multiply $- 218$ by $- 18032$ .

$44 \cdot 115002558 - 75 (20 (- 2172510 + 3930976 - 55 \cdot - 13349) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.5.1.3

Multiply $- 55$ by $- 13349$ .

$44 \cdot 115002558 - 75 (20 (- 2172510 + 3930976 + 734195) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.5.2

Add $- 2172510$ and $3930976$ .

$44 \cdot 115002558 - 75 (20 (1758466 + 734195) + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.2.5.3

Add $1758466$ and $734195$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3

Evaluate $| \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} |$ .

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Step 2.3.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.3.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.3.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3.3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.3.3.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.3.3.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.3.3.1.6

Multiply element $a_{12}$ by its cofactor.

$- 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.3.3.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.3.3.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.3.3.1.9

Add the terms together.

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} | - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.2

Evaluate $| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$ .

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Step 2.3.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 (82 \cdot 74 - 143 \cdot 106) - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.2.2

Simplify the determinant.

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Step 2.3.3.2.2.1

Simplify each term.

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Step 2.3.3.2.2.1.1

Multiply $82$ by $74$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 (6068 - 143 \cdot 106) - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.2.2.1.2

Multiply $- 143$ by $106$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 (6068 - 15158) - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.2.2.2

Subtract $15158$ from $6068$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.3

Evaluate $| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$ .

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Step 2.3.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 (119 \cdot 74 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.3.2

Simplify the determinant.

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Step 2.3.3.3.2.1

Simplify each term.

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Step 2.3.3.3.2.1.1

Multiply $119$ by $74$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 (8806 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.3.2.1.2

Multiply $- 162$ by $106$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 (8806 - 17172) - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.3.2.2

Subtract $17172$ from $8806$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.4

Evaluate $| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$ .

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Step 2.3.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 (119 \cdot 143 - 162 \cdot 82)) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.4.2

Simplify the determinant.

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Step 2.3.3.4.2.1

Simplify each term.

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Step 2.3.3.4.2.1.1

Multiply $119$ by $143$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 (17017 - 162 \cdot 82)) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.4.2.1.2

Multiply $- 162$ by $82$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 (17017 - 13284)) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.4.2.2

Subtract $13284$ from $17017$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 \cdot 3733) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.5

Simplify the determinant.

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Step 2.3.3.5.1

Simplify each term.

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Step 2.3.3.5.1.1

Multiply $113$ by $- 9090$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (- 1027170 - 218 \cdot - 8366 - 55 \cdot 3733) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.5.1.2

Multiply $- 218$ by $- 8366$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (- 1027170 + 1823788 - 55 \cdot 3733) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.5.1.3

Multiply $- 55$ by $3733$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (- 1027170 + 1823788 - 205315) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.5.2

Add $- 1027170$ and $1823788$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 (796618 - 205315) + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.3.5.3

Subtract $205315$ from $796618$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4

Evaluate $| \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} |$ .

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Step 2.3.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.3.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.3.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.3.4.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.3.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.3.4.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.3.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.3.4.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.3.4.1.9

Add the terms together.

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.2

Evaluate $| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$ .

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Step 2.3.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 (7 \cdot 74 - 175 \cdot 106) - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.2.2

Simplify the determinant.

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Step 2.3.4.2.2.1

Simplify each term.

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Step 2.3.4.2.2.1.1

Multiply $7$ by $74$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 (518 - 175 \cdot 106) - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.2.2.1.2

Multiply $- 175$ by $106$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 (518 - 18550) - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.2.2.2

Subtract $18550$ from $518$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.3

Evaluate $| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$ .

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Step 2.3.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 (119 \cdot 74 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.3.2

Simplify the determinant.

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Step 2.3.4.3.2.1

Simplify each term.

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Step 2.3.4.3.2.1.1

Multiply $119$ by $74$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 (8806 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.3.2.1.2

Multiply $- 162$ by $106$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 (8806 - 17172) - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.3.2.2

Subtract $17172$ from $8806$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.4

Evaluate $| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$ .

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Step 2.3.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 (119 \cdot 175 - 162 \cdot 7)) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.4.2

Simplify the determinant.

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Step 2.3.4.4.2.1

Simplify each term.

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Step 2.3.4.4.2.1.1

Multiply $119$ by $175$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 (20825 - 162 \cdot 7)) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.4.2.1.2

Multiply $- 162$ by $7$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 (20825 - 1134)) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.4.2.2

Subtract $1134$ from $20825$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 \cdot 19691) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.5

Simplify the determinant.

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Step 2.3.4.5.1

Simplify each term.

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Step 2.3.4.5.1.1

Multiply $113$ by $- 18032$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (- 2037616 - 239 \cdot - 8366 - 55 \cdot 19691) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.5.1.2

Multiply $- 239$ by $- 8366$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (- 2037616 + 1999474 - 55 \cdot 19691) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.5.1.3

Multiply $- 55$ by $19691$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (- 2037616 + 1999474 - 1083005) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.5.2

Add $- 2037616$ and $1999474$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 (- 38142 - 1083005) - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.4.5.3

Subtract $1083005$ from $- 38142$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5

Evaluate $| \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |$ .

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Step 2.3.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.3.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.3.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.3.5.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.3.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.3.5.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.3.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.3.5.1.8

Multiply element $a_{13}$ by its cofactor.

$218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.3.5.1.9

Add the terms together.

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} | - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.2

Evaluate $| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$ .

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Step 2.3.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 (7 \cdot 143 - 175 \cdot 82) - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.2.2

Simplify the determinant.

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Step 2.3.5.2.2.1

Simplify each term.

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Step 2.3.5.2.2.1.1

Multiply $7$ by $143$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 (1001 - 175 \cdot 82) - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.2.2.1.2

Multiply $- 175$ by $82$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 (1001 - 14350) - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.2.2.2

Subtract $14350$ from $1001$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.3

Evaluate $| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$ .

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Step 2.3.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 (119 \cdot 143 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.3.2

Simplify the determinant.

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Step 2.3.5.3.2.1

Simplify each term.

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Step 2.3.5.3.2.1.1

Multiply $119$ by $143$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 (17017 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.3.2.1.2

Multiply $- 162$ by $82$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 (17017 - 13284) + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.3.2.2

Subtract $13284$ from $17017$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 \cdot 3733 + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.4

Evaluate $| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$ .

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Step 2.3.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 \cdot 3733 + 218 (119 \cdot 175 - 162 \cdot 7))) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.4.2

Simplify the determinant.

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Step 2.3.5.4.2.1

Simplify each term.

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Step 2.3.5.4.2.1.1

Multiply $119$ by $175$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 \cdot 3733 + 218 (20825 - 162 \cdot 7))) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.4.2.1.2

Multiply $- 162$ by $7$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 \cdot 3733 + 218 (20825 - 1134))) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.4.2.2

Subtract $1134$ from $20825$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (113 \cdot - 13349 - 239 \cdot 3733 + 218 \cdot 19691)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.5

Simplify the determinant.

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Step 2.3.5.5.1

Simplify each term.

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Step 2.3.5.5.1.1

Multiply $113$ by $- 13349$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (- 1508437 - 239 \cdot 3733 + 218 \cdot 19691)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.5.1.2

Multiply $- 239$ by $3733$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (- 1508437 - 892187 + 218 \cdot 19691)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.5.1.3

Multiply $218$ by $19691$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (- 1508437 - 892187 + 4292638)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.5.2

Subtract $892187$ from $- 1508437$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 (- 2400624 + 4292638)) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.5.5.3

Add $- 2400624$ and $4292638$ .

$44 \cdot 115002558 - 75 (20 \cdot 2492661 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 \cdot 1892014) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6

Simplify the determinant.

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Step 2.3.6.1

Simplify each term.

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Step 2.3.6.1.1

Multiply $20$ by $2492661$ .

$44 \cdot 115002558 - 75 (49853220 + 164 \cdot 591303 + 18 \cdot - 1121147 - 105 \cdot 1892014) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6.1.2

Multiply $164$ by $591303$ .

$44 \cdot 115002558 - 75 (49853220 + 96973692 + 18 \cdot - 1121147 - 105 \cdot 1892014) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6.1.3

Multiply $18$ by $- 1121147$ .

$44 \cdot 115002558 - 75 (49853220 + 96973692 - 20180646 - 105 \cdot 1892014) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6.1.4

Multiply $- 105$ by $1892014$ .

$44 \cdot 115002558 - 75 (49853220 + 96973692 - 20180646 - 198661470) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6.2

Add $49853220$ and $96973692$ .

$44 \cdot 115002558 - 75 (146826912 - 20180646 - 198661470) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6.3

Subtract $20180646$ from $146826912$ .

$44 \cdot 115002558 - 75 (126646266 - 198661470) - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.3.6.4

Subtract $198661470$ from $126646266$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 | \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} | + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4

Evaluate $| \begin{array}{c} 20 & 169 & 18 & 105 \\ 113 & - 82 & 218 & - 55 \\ 119 & 83 & 82 & 106 \\ 162 & 20 & 143 & 74 \end{array} |$ .

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Step 2.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 2.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} |$

Step 2.4.1.4

Multiply element $a_{11}$ by its cofactor.

$20 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} |$

Step 2.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} |$

Step 2.4.1.6

Multiply element $a_{12}$ by its cofactor.

$- 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} |$

Step 2.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} |$

Step 2.4.1.8

Multiply element $a_{13}$ by its cofactor.

$18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} |$

Step 2.4.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |$

Step 2.4.1.10

Multiply element $a_{14}$ by its cofactor.

$- 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |$

Step 2.4.1.11

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 | \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} | - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2

Evaluate $| \begin{array}{c} - 82 & 218 & - 55 \\ 83 & 82 & 106 \\ 20 & 143 & 74 \end{array} |$ .

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Step 2.4.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.4.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.4.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.4.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.4.2.1.4

Multiply element $a_{11}$ by its cofactor.

$- 82 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.4.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.4.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.4.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.4.2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.4.2.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} | - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.2

Evaluate $| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$ .

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Step 2.4.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 (82 \cdot 74 - 143 \cdot 106) - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.2.2

Simplify the determinant.

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Step 2.4.2.2.2.1

Simplify each term.

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Step 2.4.2.2.2.1.1

Multiply $82$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 (6068 - 143 \cdot 106) - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.2.2.1.2

Multiply $- 143$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 (6068 - 15158) - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.2.2.2

Subtract $15158$ from $6068$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.3

Evaluate $| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$ .

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Step 2.4.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 (83 \cdot 74 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.3.2

Simplify the determinant.

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Step 2.4.2.3.2.1

Simplify each term.

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Step 2.4.2.3.2.1.1

Multiply $83$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 (6142 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.3.2.1.2

Multiply $- 20$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 (6142 - 2120) - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.3.2.2

Subtract $2120$ from $6142$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.4

Evaluate $| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$ .

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Step 2.4.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 (83 \cdot 143 - 20 \cdot 82)) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.4.2

Simplify the determinant.

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Step 2.4.2.4.2.1

Simplify each term.

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Step 2.4.2.4.2.1.1

Multiply $83$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 (11869 - 20 \cdot 82)) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.4.2.1.2

Multiply $- 20$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 (11869 - 1640)) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.4.2.2

Subtract $1640$ from $11869$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 82 \cdot - 9090 - 218 \cdot 4022 - 55 \cdot 10229) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.5

Simplify the determinant.

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Step 2.4.2.5.1

Simplify each term.

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Step 2.4.2.5.1.1

Multiply $- 82$ by $- 9090$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (745380 - 218 \cdot 4022 - 55 \cdot 10229) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.5.1.2

Multiply $- 218$ by $4022$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (745380 - 876796 - 55 \cdot 10229) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.5.1.3

Multiply $- 55$ by $10229$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (745380 - 876796 - 562595) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.5.2

Subtract $876796$ from $745380$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 (- 131416 - 562595) - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.2.5.3

Subtract $562595$ from $- 131416$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 | \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} | + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3

Evaluate $| \begin{array}{c} 113 & 218 & - 55 \\ 119 & 82 & 106 \\ 162 & 143 & 74 \end{array} |$ .

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Step 2.4.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.4.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.4.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.4.3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.4.3.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$

Step 2.4.3.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.4.3.1.6

Multiply element $a_{12}$ by its cofactor.

$- 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.4.3.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.4.3.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.4.3.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 | \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} | - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.2

Evaluate $| \begin{array}{c} 82 & 106 \\ 143 & 74 \end{array} |$ .

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Step 2.4.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 (82 \cdot 74 - 143 \cdot 106) - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.2.2

Simplify the determinant.

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Step 2.4.3.2.2.1

Simplify each term.

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Step 2.4.3.2.2.1.1

Multiply $82$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 (6068 - 143 \cdot 106) - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.2.2.1.2

Multiply $- 143$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 (6068 - 15158) - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.2.2.2

Subtract $15158$ from $6068$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.3

Evaluate $| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$ .

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Step 2.4.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 (119 \cdot 74 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.3.2

Simplify the determinant.

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Step 2.4.3.3.2.1

Simplify each term.

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Step 2.4.3.3.2.1.1

Multiply $119$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 (8806 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.3.2.1.2

Multiply $- 162$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 (8806 - 17172) - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.3.2.2

Subtract $17172$ from $8806$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.4

Evaluate $| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$ .

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Step 2.4.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 (119 \cdot 143 - 162 \cdot 82)) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.4.2

Simplify the determinant.

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Step 2.4.3.4.2.1

Simplify each term.

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Step 2.4.3.4.2.1.1

Multiply $119$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 (17017 - 162 \cdot 82)) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.4.2.1.2

Multiply $- 162$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 (17017 - 13284)) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.4.2.2

Subtract $13284$ from $17017$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (113 \cdot - 9090 - 218 \cdot - 8366 - 55 \cdot 3733) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.5

Simplify the determinant.

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Step 2.4.3.5.1

Simplify each term.

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Step 2.4.3.5.1.1

Multiply $113$ by $- 9090$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (- 1027170 - 218 \cdot - 8366 - 55 \cdot 3733) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.5.1.2

Multiply $- 218$ by $- 8366$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (- 1027170 + 1823788 - 55 \cdot 3733) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.5.1.3

Multiply $- 55$ by $3733$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (- 1027170 + 1823788 - 205315) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.5.2

Add $- 1027170$ and $1823788$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 (796618 - 205315) + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.3.5.3

Subtract $205315$ from $796618$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4

Evaluate $| \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} |$ .

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Step 2.4.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.4.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.4.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.4.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.4.4.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.4.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.4.4.1.6

Multiply element $a_{12}$ by its cofactor.

$82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.4.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.4.4.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.4.4.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.2

Evaluate $| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$ .

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Step 2.4.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 (83 \cdot 74 - 20 \cdot 106) + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.2.2

Simplify the determinant.

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Step 2.4.4.2.2.1

Simplify each term.

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Step 2.4.4.2.2.1.1

Multiply $83$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 (6142 - 20 \cdot 106) + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.2.2.1.2

Multiply $- 20$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 (6142 - 2120) + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.2.2.2

Subtract $2120$ from $6142$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.3

Evaluate $| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$ .

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Step 2.4.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 (119 \cdot 74 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.3.2

Simplify the determinant.

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Step 2.4.4.3.2.1

Simplify each term.

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Step 2.4.4.3.2.1.1

Multiply $119$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 (8806 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.3.2.1.2

Multiply $- 162$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 (8806 - 17172) - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.3.2.2

Subtract $17172$ from $8806$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 \cdot - 8366 - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.4

Evaluate $| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$ .

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Step 2.4.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 \cdot - 8366 - 55 (119 \cdot 20 - 162 \cdot 83)) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.4.2

Simplify the determinant.

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Step 2.4.4.4.2.1

Simplify each term.

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Step 2.4.4.4.2.1.1

Multiply $119$ by $20$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 \cdot - 8366 - 55 (2380 - 162 \cdot 83)) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.4.2.1.2

Multiply $- 162$ by $83$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 \cdot - 8366 - 55 (2380 - 13446)) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.4.2.2

Subtract $13446$ from $2380$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (113 \cdot 4022 + 82 \cdot - 8366 - 55 \cdot - 11066) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.5

Simplify the determinant.

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Step 2.4.4.5.1

Simplify each term.

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Step 2.4.4.5.1.1

Multiply $113$ by $4022$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (454486 + 82 \cdot - 8366 - 55 \cdot - 11066) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.5.1.2

Multiply $82$ by $- 8366$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (454486 - 686012 - 55 \cdot - 11066) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.5.1.3

Multiply $- 55$ by $- 11066$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (454486 - 686012 + 608630) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.5.2

Subtract $686012$ from $454486$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 (- 231526 + 608630) - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.4.5.3

Add $- 231526$ and $608630$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5

Evaluate $| \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |$ .

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Step 2.4.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.4.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.4.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.4.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.4.5.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.4.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.4.5.1.6

Multiply element $a_{12}$ by its cofactor.

$82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.4.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.4.5.1.8

Multiply element $a_{13}$ by its cofactor.

$218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.4.5.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.2

Evaluate $| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$ .

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Step 2.4.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 (83 \cdot 143 - 20 \cdot 82) + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.2.2

Simplify the determinant.

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Step 2.4.5.2.2.1

Simplify each term.

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Step 2.4.5.2.2.1.1

Multiply $83$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 (11869 - 20 \cdot 82) + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.2.2.1.2

Multiply $- 20$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 (11869 - 1640) + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.2.2.2

Subtract $1640$ from $11869$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.3

Evaluate $| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$ .

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Step 2.4.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 (119 \cdot 143 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.3.2

Simplify the determinant.

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Step 2.4.5.3.2.1

Simplify each term.

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Step 2.4.5.3.2.1.1

Multiply $119$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 (17017 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.3.2.1.2

Multiply $- 162$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 (17017 - 13284) + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.3.2.2

Subtract $13284$ from $17017$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 \cdot 3733 + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.4

Evaluate $| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$ .

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Step 2.4.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 \cdot 3733 + 218 (119 \cdot 20 - 162 \cdot 83))) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.4.2

Simplify the determinant.

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Step 2.4.5.4.2.1

Simplify each term.

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Step 2.4.5.4.2.1.1

Multiply $119$ by $20$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 \cdot 3733 + 218 (2380 - 162 \cdot 83))) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.4.2.1.2

Multiply $- 162$ by $83$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 \cdot 3733 + 218 (2380 - 13446))) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.4.2.2

Subtract $13446$ from $2380$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (113 \cdot 10229 + 82 \cdot 3733 + 218 \cdot - 11066)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.5

Simplify the determinant.

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Step 2.4.5.5.1

Simplify each term.

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Step 2.4.5.5.1.1

Multiply $113$ by $10229$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (1155877 + 82 \cdot 3733 + 218 \cdot - 11066)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.5.1.2

Multiply $82$ by $3733$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (1155877 + 306106 + 218 \cdot - 11066)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.5.1.3

Multiply $218$ by $- 11066$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (1155877 + 306106 - 2412388)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.5.2

Add $1155877$ and $306106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 (1461983 - 2412388)) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.5.5.3

Subtract $2412388$ from $1461983$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (20 \cdot - 694011 - 169 \cdot 591303 + 18 \cdot 377104 - 105 \cdot - 950405) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6

Simplify the determinant.

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Step 2.4.6.1

Simplify each term.

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Step 2.4.6.1.1

Multiply $20$ by $- 694011$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (- 13880220 - 169 \cdot 591303 + 18 \cdot 377104 - 105 \cdot - 950405) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6.1.2

Multiply $- 169$ by $591303$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (- 13880220 - 99930207 + 18 \cdot 377104 - 105 \cdot - 950405) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6.1.3

Multiply $18$ by $377104$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (- 13880220 - 99930207 + 6787872 - 105 \cdot - 950405) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6.1.4

Multiply $- 105$ by $- 950405$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (- 13880220 - 99930207 + 6787872 + 99792525) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6.2

Subtract $99930207$ from $- 13880220$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (- 113810427 + 6787872 + 99792525) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6.3

Add $- 113810427$ and $6787872$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 (- 107022555 + 99792525) + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.4.6.4

Add $- 107022555$ and $99792525$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 | \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} | + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5

Evaluate $| \begin{array}{c} 20 & 169 & - 164 & 105 \\ 113 & - 82 & 239 & - 55 \\ 119 & 83 & 7 & 106 \\ 162 & 20 & 175 & 74 \end{array} |$ .

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Step 2.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 2.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} |$

Step 2.5.1.4

Multiply element $a_{11}$ by its cofactor.

$20 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} |$

Step 2.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} |$

Step 2.5.1.6

Multiply element $a_{12}$ by its cofactor.

$- 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} |$

Step 2.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} |$

Step 2.5.1.8

Multiply element $a_{13}$ by its cofactor.

$- 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} |$

Step 2.5.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |$

Step 2.5.1.10

Multiply element $a_{14}$ by its cofactor.

$- 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |$

Step 2.5.1.11

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 | \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} | - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2

Evaluate $| \begin{array}{c} - 82 & 239 & - 55 \\ 83 & 7 & 106 \\ 20 & 175 & 74 \end{array} |$ .

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Step 2.5.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.5.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.5.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.5.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.5.2.1.4

Multiply element $a_{11}$ by its cofactor.

$- 82 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.5.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.5.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.5.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.5.2.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.5.2.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.2

Evaluate $| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$ .

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Step 2.5.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 (7 \cdot 74 - 175 \cdot 106) - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.2.2

Simplify the determinant.

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Step 2.5.2.2.2.1

Simplify each term.

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Step 2.5.2.2.2.1.1

Multiply $7$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 (518 - 175 \cdot 106) - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.2.2.1.2

Multiply $- 175$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 (518 - 18550) - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.2.2.2

Subtract $18550$ from $518$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.3

Evaluate $| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$ .

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Step 2.5.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 (83 \cdot 74 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.3.2

Simplify the determinant.

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Step 2.5.2.3.2.1

Simplify each term.

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Step 2.5.2.3.2.1.1

Multiply $83$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 (6142 - 20 \cdot 106) - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.3.2.1.2

Multiply $- 20$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 (6142 - 2120) - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.3.2.2

Subtract $2120$ from $6142$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.4

Evaluate $| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$ .

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Step 2.5.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 (83 \cdot 175 - 20 \cdot 7)) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.4.2

Simplify the determinant.

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Step 2.5.2.4.2.1

Simplify each term.

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Step 2.5.2.4.2.1.1

Multiply $83$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 (14525 - 20 \cdot 7)) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.4.2.1.2

Multiply $- 20$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 (14525 - 140)) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.4.2.2

Subtract $140$ from $14525$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (- 82 \cdot - 18032 - 239 \cdot 4022 - 55 \cdot 14385) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.5

Simplify the determinant.

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Step 2.5.2.5.1

Simplify each term.

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Step 2.5.2.5.1.1

Multiply $- 82$ by $- 18032$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (1478624 - 239 \cdot 4022 - 55 \cdot 14385) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.5.1.2

Multiply $- 239$ by $4022$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (1478624 - 961258 - 55 \cdot 14385) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.5.1.3

Multiply $- 55$ by $14385$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (1478624 - 961258 - 791175) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.5.2

Subtract $961258$ from $1478624$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 (517366 - 791175) - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.2.5.3

Subtract $791175$ from $517366$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 | \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3

Evaluate $| \begin{array}{c} 113 & 239 & - 55 \\ 119 & 7 & 106 \\ 162 & 175 & 74 \end{array} |$ .

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Step 2.5.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.5.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.5.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.5.3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.5.3.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$

Step 2.5.3.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.5.3.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.5.3.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.5.3.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.5.3.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 | \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} | - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.2

Evaluate $| \begin{array}{c} 7 & 106 \\ 175 & 74 \end{array} |$ .

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Step 2.5.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 (7 \cdot 74 - 175 \cdot 106) - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.2.2

Simplify the determinant.

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Step 2.5.3.2.2.1

Simplify each term.

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Step 2.5.3.2.2.1.1

Multiply $7$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 (518 - 175 \cdot 106) - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.2.2.1.2

Multiply $- 175$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 (518 - 18550) - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.2.2.2

Subtract $18550$ from $518$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.3

Evaluate $| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$ .

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Step 2.5.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 (119 \cdot 74 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.3.2

Simplify the determinant.

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Step 2.5.3.3.2.1

Simplify each term.

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Step 2.5.3.3.2.1.1

Multiply $119$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 (8806 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.3.2.1.2

Multiply $- 162$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 (8806 - 17172) - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.3.2.2

Subtract $17172$ from $8806$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.4

Evaluate $| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$ .

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Step 2.5.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 (119 \cdot 175 - 162 \cdot 7)) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.4.2

Simplify the determinant.

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Step 2.5.3.4.2.1

Simplify each term.

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Step 2.5.3.4.2.1.1

Multiply $119$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 (20825 - 162 \cdot 7)) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.4.2.1.2

Multiply $- 162$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 (20825 - 1134)) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.4.2.2

Subtract $1134$ from $20825$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (113 \cdot - 18032 - 239 \cdot - 8366 - 55 \cdot 19691) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.5

Simplify the determinant.

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Step 2.5.3.5.1

Simplify each term.

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Step 2.5.3.5.1.1

Multiply $113$ by $- 18032$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (- 2037616 - 239 \cdot - 8366 - 55 \cdot 19691) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.5.1.2

Multiply $- 239$ by $- 8366$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (- 2037616 + 1999474 - 55 \cdot 19691) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.5.1.3

Multiply $- 55$ by $19691$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (- 2037616 + 1999474 - 1083005) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.5.2

Add $- 2037616$ and $1999474$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 (- 38142 - 1083005) - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.3.5.3

Subtract $1083005$ from $- 38142$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 | \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} | - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4

Evaluate $| \begin{array}{c} 113 & - 82 & - 55 \\ 119 & 83 & 106 \\ 162 & 20 & 74 \end{array} |$ .

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Step 2.5.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.5.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.5.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.5.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.5.4.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$

Step 2.5.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.5.4.1.6

Multiply element $a_{12}$ by its cofactor.

$82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$

Step 2.5.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.5.4.1.8

Multiply element $a_{13}$ by its cofactor.

$- 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.5.4.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 | \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} | + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.2

Evaluate $| \begin{array}{c} 83 & 106 \\ 20 & 74 \end{array} |$ .

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Step 2.5.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 (83 \cdot 74 - 20 \cdot 106) + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.2.2

Simplify the determinant.

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Step 2.5.4.2.2.1

Simplify each term.

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Step 2.5.4.2.2.1.1

Multiply $83$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 (6142 - 20 \cdot 106) + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.2.2.1.2

Multiply $- 20$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 (6142 - 2120) + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.2.2.2

Subtract $2120$ from $6142$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 | \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} | - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.3

Evaluate $| \begin{array}{c} 119 & 106 \\ 162 & 74 \end{array} |$ .

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Step 2.5.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 (119 \cdot 74 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.3.2

Simplify the determinant.

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Step 2.5.4.3.2.1

Simplify each term.

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Step 2.5.4.3.2.1.1

Multiply $119$ by $74$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 (8806 - 162 \cdot 106) - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.3.2.1.2

Multiply $- 162$ by $106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 (8806 - 17172) - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.3.2.2

Subtract $17172$ from $8806$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 \cdot - 8366 - 55 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.4

Evaluate $| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$ .

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Step 2.5.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 \cdot - 8366 - 55 (119 \cdot 20 - 162 \cdot 83)) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.4.2

Simplify the determinant.

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Step 2.5.4.4.2.1

Simplify each term.

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Step 2.5.4.4.2.1.1

Multiply $119$ by $20$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 \cdot - 8366 - 55 (2380 - 162 \cdot 83)) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.4.2.1.2

Multiply $- 162$ by $83$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 \cdot - 8366 - 55 (2380 - 13446)) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.4.2.2

Subtract $13446$ from $2380$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (113 \cdot 4022 + 82 \cdot - 8366 - 55 \cdot - 11066) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.5

Simplify the determinant.

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Step 2.5.4.5.1

Simplify each term.

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Step 2.5.4.5.1.1

Multiply $113$ by $4022$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (454486 + 82 \cdot - 8366 - 55 \cdot - 11066) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.5.1.2

Multiply $82$ by $- 8366$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (454486 - 686012 - 55 \cdot - 11066) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.5.1.3

Multiply $- 55$ by $- 11066$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (454486 - 686012 + 608630) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.5.2

Subtract $686012$ from $454486$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 (- 231526 + 608630) - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.4.5.3

Add $- 231526$ and $608630$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5

Evaluate $| \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |$ .

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Step 2.5.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.5.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.5.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.5.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.5.5.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.5.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.5.5.1.6

Multiply element $a_{12}$ by its cofactor.

$82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.5.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.5.5.1.8

Multiply element $a_{13}$ by its cofactor.

$239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.5.5.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} | + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.2

Evaluate $| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$ .

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Step 2.5.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 (83 \cdot 175 - 20 \cdot 7) + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.2.2

Simplify the determinant.

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Step 2.5.5.2.2.1

Simplify each term.

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Step 2.5.5.2.2.1.1

Multiply $83$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 (14525 - 20 \cdot 7) + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.2.2.1.2

Multiply $- 20$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 (14525 - 140) + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.2.2.2

Subtract $140$ from $14525$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.3

Evaluate $| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$ .

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Step 2.5.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 (119 \cdot 175 - 162 \cdot 7) + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.3.2

Simplify the determinant.

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Step 2.5.5.3.2.1

Simplify each term.

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Step 2.5.5.3.2.1.1

Multiply $119$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 (20825 - 162 \cdot 7) + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.3.2.1.2

Multiply $- 162$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 (20825 - 1134) + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.3.2.2

Subtract $1134$ from $20825$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 \cdot 19691 + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.4

Evaluate $| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$ .

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Step 2.5.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 \cdot 19691 + 239 (119 \cdot 20 - 162 \cdot 83))) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.4.2

Simplify the determinant.

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Step 2.5.5.4.2.1

Simplify each term.

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Step 2.5.5.4.2.1.1

Multiply $119$ by $20$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 \cdot 19691 + 239 (2380 - 162 \cdot 83))) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.4.2.1.2

Multiply $- 162$ by $83$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 \cdot 19691 + 239 (2380 - 13446))) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.4.2.2

Subtract $13446$ from $2380$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (113 \cdot 14385 + 82 \cdot 19691 + 239 \cdot - 11066)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.5

Simplify the determinant.

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Step 2.5.5.5.1

Simplify each term.

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Step 2.5.5.5.1.1

Multiply $113$ by $14385$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (1625505 + 82 \cdot 19691 + 239 \cdot - 11066)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.5.1.2

Multiply $82$ by $19691$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (1625505 + 1614662 + 239 \cdot - 11066)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.5.1.3

Multiply $239$ by $- 11066$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (1625505 + 1614662 - 2644774)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.5.2

Add $1625505$ and $1614662$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 (3240167 - 2644774)) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.5.5.3

Subtract $2644774$ from $3240167$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (20 \cdot - 273809 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 \cdot 595393) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6

Simplify the determinant.

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Step 2.5.6.1

Simplify each term.

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Step 2.5.6.1.1

Multiply $20$ by $- 273809$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (- 5476180 - 169 \cdot - 1121147 - 164 \cdot 377104 - 105 \cdot 595393) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6.1.2

Multiply $- 169$ by $- 1121147$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (- 5476180 + 189473843 - 164 \cdot 377104 - 105 \cdot 595393) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6.1.3

Multiply $- 164$ by $377104$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (- 5476180 + 189473843 - 61845056 - 105 \cdot 595393) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6.1.4

Multiply $- 105$ by $595393$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (- 5476180 + 189473843 - 61845056 - 62516265) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6.2

Add $- 5476180$ and $189473843$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (183997663 - 61845056 - 62516265) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6.3

Subtract $61845056$ from $183997663$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 (122152607 - 62516265) + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.5.6.4

Subtract $62516265$ from $122152607$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 | \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$

Step 2.6

Evaluate $| \begin{array}{c} 20 & 169 & - 164 & 18 \\ 113 & - 82 & 239 & 218 \\ 119 & 83 & 7 & 82 \\ 162 & 20 & 175 & 143 \end{array} |$ .

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Step 2.6.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.6.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 2.6.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.6.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |$

Step 2.6.1.4

Multiply element $a_{11}$ by its cofactor.

$20 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |$

Step 2.6.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |$

Step 2.6.1.6

Multiply element $a_{12}$ by its cofactor.

$- 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |$

Step 2.6.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |$

Step 2.6.1.8

Multiply element $a_{13}$ by its cofactor.

$- 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |$

Step 2.6.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |$

Step 2.6.1.10

Multiply element $a_{14}$ by its cofactor.

$- 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |$

Step 2.6.1.11

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 | \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} | - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2

Evaluate $| \begin{array}{c} - 82 & 239 & 218 \\ 83 & 7 & 82 \\ 20 & 175 & 143 \end{array} |$ .

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Step 2.6.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.6.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.6.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.6.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.6.2.1.4

Multiply element $a_{11}$ by its cofactor.

$- 82 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.6.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.6.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.6.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.6.2.1.8

Multiply element $a_{13}$ by its cofactor.

$218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.6.2.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} | - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.2

Evaluate $| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$ .

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Step 2.6.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 (7 \cdot 143 - 175 \cdot 82) - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.2.2

Simplify the determinant.

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Step 2.6.2.2.2.1

Simplify each term.

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Step 2.6.2.2.2.1.1

Multiply $7$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 (1001 - 175 \cdot 82) - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.2.2.1.2

Multiply $- 175$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 (1001 - 14350) - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.2.2.2

Subtract $14350$ from $1001$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.3

Evaluate $| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$ .

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Step 2.6.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 (83 \cdot 143 - 20 \cdot 82) + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.3.2

Simplify the determinant.

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Step 2.6.2.3.2.1

Simplify each term.

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Step 2.6.2.3.2.1.1

Multiply $83$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 (11869 - 20 \cdot 82) + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.3.2.1.2

Multiply $- 20$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 (11869 - 1640) + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.3.2.2

Subtract $1640$ from $11869$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.4

Evaluate $| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$ .

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Step 2.6.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 (83 \cdot 175 - 20 \cdot 7)) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.4.2

Simplify the determinant.

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Step 2.6.2.4.2.1

Simplify each term.

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Step 2.6.2.4.2.1.1

Multiply $83$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 (14525 - 20 \cdot 7)) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.4.2.1.2

Multiply $- 20$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 (14525 - 140)) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.4.2.2

Subtract $140$ from $14525$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 82 \cdot - 13349 - 239 \cdot 10229 + 218 \cdot 14385) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.5

Simplify the determinant.

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Step 2.6.2.5.1

Simplify each term.

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Step 2.6.2.5.1.1

Multiply $- 82$ by $- 13349$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (1094618 - 239 \cdot 10229 + 218 \cdot 14385) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.5.1.2

Multiply $- 239$ by $10229$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (1094618 - 2444731 + 218 \cdot 14385) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.5.1.3

Multiply $218$ by $14385$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (1094618 - 2444731 + 3135930) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.5.2

Subtract $2444731$ from $1094618$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 (- 1350113 + 3135930) - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.2.5.3

Add $- 1350113$ and $3135930$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 | \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} | - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3

Evaluate $| \begin{array}{c} 113 & 239 & 218 \\ 119 & 7 & 82 \\ 162 & 175 & 143 \end{array} |$ .

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Step 2.6.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.6.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.6.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.6.3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.6.3.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$

Step 2.6.3.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.6.3.1.6

Multiply element $a_{12}$ by its cofactor.

$- 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.6.3.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.6.3.1.8

Multiply element $a_{13}$ by its cofactor.

$218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.6.3.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 | \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} | - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.2

Evaluate $| \begin{array}{c} 7 & 82 \\ 175 & 143 \end{array} |$ .

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Step 2.6.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 (7 \cdot 143 - 175 \cdot 82) - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.2.2

Simplify the determinant.

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Step 2.6.3.2.2.1

Simplify each term.

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Step 2.6.3.2.2.1.1

Multiply $7$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 (1001 - 175 \cdot 82) - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.2.2.1.2

Multiply $- 175$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 (1001 - 14350) - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.2.2.2

Subtract $14350$ from $1001$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.3

Evaluate $| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$ .

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Step 2.6.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 (119 \cdot 143 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.3.2

Simplify the determinant.

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Step 2.6.3.3.2.1

Simplify each term.

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Step 2.6.3.3.2.1.1

Multiply $119$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 (17017 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.3.2.1.2

Multiply $- 162$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 (17017 - 13284) + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.3.2.2

Subtract $13284$ from $17017$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 \cdot 3733 + 218 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.4

Evaluate $| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$ .

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Step 2.6.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 \cdot 3733 + 218 (119 \cdot 175 - 162 \cdot 7)) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.4.2

Simplify the determinant.

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Step 2.6.3.4.2.1

Simplify each term.

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Step 2.6.3.4.2.1.1

Multiply $119$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 \cdot 3733 + 218 (20825 - 162 \cdot 7)) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.4.2.1.2

Multiply $- 162$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 \cdot 3733 + 218 (20825 - 1134)) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.4.2.2

Subtract $1134$ from $20825$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (113 \cdot - 13349 - 239 \cdot 3733 + 218 \cdot 19691) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.5

Simplify the determinant.

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Step 2.6.3.5.1

Simplify each term.

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Step 2.6.3.5.1.1

Multiply $113$ by $- 13349$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (- 1508437 - 239 \cdot 3733 + 218 \cdot 19691) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.5.1.2

Multiply $- 239$ by $3733$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (- 1508437 - 892187 + 218 \cdot 19691) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.5.1.3

Multiply $218$ by $19691$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (- 1508437 - 892187 + 4292638) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.5.2

Subtract $892187$ from $- 1508437$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 (- 2400624 + 4292638) - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.3.5.3

Add $- 2400624$ and $4292638$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 | \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} | - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4

Evaluate $| \begin{array}{c} 113 & - 82 & 218 \\ 119 & 83 & 82 \\ 162 & 20 & 143 \end{array} |$ .

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Step 2.6.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.6.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.6.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.6.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.6.4.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$

Step 2.6.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.6.4.1.6

Multiply element $a_{12}$ by its cofactor.

$82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$

Step 2.6.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.6.4.1.8

Multiply element $a_{13}$ by its cofactor.

$218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.6.4.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 | \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} | + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.2

Evaluate $| \begin{array}{c} 83 & 82 \\ 20 & 143 \end{array} |$ .

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Step 2.6.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 (83 \cdot 143 - 20 \cdot 82) + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.2.2

Simplify the determinant.

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Step 2.6.4.2.2.1

Simplify each term.

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Step 2.6.4.2.2.1.1

Multiply $83$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 (11869 - 20 \cdot 82) + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.2.2.1.2

Multiply $- 20$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 (11869 - 1640) + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.2.2.2

Subtract $1640$ from $11869$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 | \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} | + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.3

Evaluate $| \begin{array}{c} 119 & 82 \\ 162 & 143 \end{array} |$ .

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Step 2.6.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 (119 \cdot 143 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.3.2

Simplify the determinant.

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Step 2.6.4.3.2.1

Simplify each term.

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Step 2.6.4.3.2.1.1

Multiply $119$ by $143$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 (17017 - 162 \cdot 82) + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.3.2.1.2

Multiply $- 162$ by $82$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 (17017 - 13284) + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.3.2.2

Subtract $13284$ from $17017$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 \cdot 3733 + 218 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.4

Evaluate $| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$ .

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Step 2.6.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 \cdot 3733 + 218 (119 \cdot 20 - 162 \cdot 83)) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.4.2

Simplify the determinant.

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Step 2.6.4.4.2.1

Simplify each term.

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Step 2.6.4.4.2.1.1

Multiply $119$ by $20$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 \cdot 3733 + 218 (2380 - 162 \cdot 83)) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.4.2.1.2

Multiply $- 162$ by $83$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 \cdot 3733 + 218 (2380 - 13446)) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.4.2.2

Subtract $13446$ from $2380$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (113 \cdot 10229 + 82 \cdot 3733 + 218 \cdot - 11066) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.5

Simplify the determinant.

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Step 2.6.4.5.1

Simplify each term.

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Step 2.6.4.5.1.1

Multiply $113$ by $10229$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (1155877 + 82 \cdot 3733 + 218 \cdot - 11066) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.5.1.2

Multiply $82$ by $3733$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (1155877 + 306106 + 218 \cdot - 11066) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.5.1.3

Multiply $218$ by $- 11066$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (1155877 + 306106 - 2412388) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.5.2

Add $1155877$ and $306106$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 (1461983 - 2412388) - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.4.5.3

Subtract $2412388$ from $1461983$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 | \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |)$

Step 2.6.5

Evaluate $| \begin{array}{c} 113 & - 82 & 239 \\ 119 & 83 & 7 \\ 162 & 20 & 175 \end{array} |$ .

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Step 2.6.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.6.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.6.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.6.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.6.5.1.4

Multiply element $a_{11}$ by its cofactor.

$113 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$

Step 2.6.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.6.5.1.6

Multiply element $a_{12}$ by its cofactor.

$82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$

Step 2.6.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.6.5.1.8

Multiply element $a_{13}$ by its cofactor.

$239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$

Step 2.6.5.1.9

Add the terms together.

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 | \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} | + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.2

Evaluate $| \begin{array}{c} 83 & 7 \\ 20 & 175 \end{array} |$ .

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Step 2.6.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 (83 \cdot 175 - 20 \cdot 7) + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.2.2

Simplify the determinant.

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Step 2.6.5.2.2.1

Simplify each term.

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Step 2.6.5.2.2.1.1

Multiply $83$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 (14525 - 20 \cdot 7) + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.2.2.1.2

Multiply $- 20$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 (14525 - 140) + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.2.2.2

Subtract $140$ from $14525$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 \cdot 14385 + 82 | \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} | + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.3

Evaluate $| \begin{array}{c} 119 & 7 \\ 162 & 175 \end{array} |$ .

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Step 2.6.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 \cdot 14385 + 82 (119 \cdot 175 - 162 \cdot 7) + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.3.2

Simplify the determinant.

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Step 2.6.5.3.2.1

Simplify each term.

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Step 2.6.5.3.2.1.1

Multiply $119$ by $175$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 \cdot 14385 + 82 (20825 - 162 \cdot 7) + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.3.2.1.2

Multiply $- 162$ by $7$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 \cdot 14385 + 82 (20825 - 1134) + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.3.2.2

Subtract $1134$ from $20825$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (113 \cdot 14385 + 82 \cdot 19691 + 239 | \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |))$

Step 2.6.5.4

Evaluate $| \begin{array}{c} 119 & 83 \\ 162 & 20 \end{array} |$ .

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Step 2.6.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

Step 2.6.5.4.2

Simplify the determinant.

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Step 2.6.5.4.2.1

Simplify each term.

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Step 2.6.5.4.2.1.1

Multiply $119$ by $20$ .

Step 2.6.5.4.2.1.2

Multiply $- 162$ by $83$ .

Step 2.6.5.4.2.2

Subtract $13446$ from $2380$ .

Step 2.6.5.5

Simplify the determinant.

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Step 2.6.5.5.1

Simplify each term.

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Step 2.6.5.5.1.1

Multiply $113$ by $14385$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (1625505 + 82 \cdot 19691 + 239 \cdot - 11066))$

Step 2.6.5.5.1.2

Multiply $82$ by $19691$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (1625505 + 1614662 + 239 \cdot - 11066))$

Step 2.6.5.5.1.3

Multiply $239$ by $- 11066$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (1625505 + 1614662 - 2644774))$

Step 2.6.5.5.2

Add $1625505$ and $1614662$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 (3240167 - 2644774))$

Step 2.6.5.5.3

Subtract $2644774$ from $3240167$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (20 \cdot 1785817 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 \cdot 595393)$

Step 2.6.6

Simplify the determinant.

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Step 2.6.6.1

Simplify each term.

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Step 2.6.6.1.1

Multiply $20$ by $1785817$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (35716340 - 169 \cdot 1892014 - 164 \cdot - 950405 - 18 \cdot 595393)$

Step 2.6.6.1.2

Multiply $- 169$ by $1892014$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (35716340 - 319750366 - 164 \cdot - 950405 - 18 \cdot 595393)$

Step 2.6.6.1.3

Multiply $- 164$ by $- 950405$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (35716340 - 319750366 + 155866420 - 18 \cdot 595393)$

Step 2.6.6.1.4

Multiply $- 18$ by $595393$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (35716340 - 319750366 + 155866420 - 10717074)$

Step 2.6.6.2

Subtract $319750366$ from $35716340$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (- 284034026 + 155866420 - 10717074)$

Step 2.6.6.3

Add $- 284034026$ and $155866420$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 (- 128167606 - 10717074)$

Step 2.6.6.4

Subtract $10717074$ from $- 128167606$ .

$44 \cdot 115002558 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 \cdot - 138884680$

Step 2.7

Simplify the determinant.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $44$ by $115002558$ .

$5060112552 - 75 \cdot - 72015204 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 \cdot - 138884680$

Step 2.7.1.2

Multiply $- 75$ by $- 72015204$ .

$5060112552 + 5401140300 - 60 \cdot - 7230030 + 33 \cdot 59636342 + 11 \cdot - 138884680$

Step 2.7.1.3

Multiply $- 60$ by $- 7230030$ .

$5060112552 + 5401140300 + 433801800 + 33 \cdot 59636342 + 11 \cdot - 138884680$

Step 2.7.1.4

Multiply $33$ by $59636342$ .

$5060112552 + 5401140300 + 433801800 + 1967999286 + 11 \cdot - 138884680$

Step 2.7.1.5

Multiply $11$ by $- 138884680$ .

$5060112552 + 5401140300 + 433801800 + 1967999286 - 1527731480$

Step 2.7.2

Add $5060112552$ and $5401140300$ .

$10461252852 + 433801800 + 1967999286 - 1527731480$

Step 2.7.3

Add $10461252852$ and $433801800$ .

$10895054652 + 1967999286 - 1527731480$

Step 2.7.4

Add $10895054652$ and $1967999286$ .

$12863053938 - 1527731480$

Step 2.7.5

Subtract $1527731480$ from $12863053938$ .

$11335322458$

Step 3

Since the determinant is non-zero, the inverse exists.

Step 4

Set up a $5 \times 10$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} 44 & 75 & - 60 & - 33 & 11 & 1 & 0 & 0 & 0 & 0 \\ 20 & 169 & - 164 & 18 & 105 & 0 & 1 & 0 & 0 & 0 \\ 113 & - 82 & 239 & 218 & - 55 & 0 & 0 & 1 & 0 & 0 \\ 119 & 83 & 7 & 82 & 106 & 0 & 0 & 0 & 1 & 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5

Find the reduced row echelon form.

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Step 5.1

Multiply each element of $R_{1}$ by $\frac{1}{44}$ to make the entry at $1, 1$ a $1$ .

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Step 5.1.1

Multiply each element of $R_{1}$ by $\frac{1}{44}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{44}{44} & \frac{75}{44} & \frac{- 60}{44} & \frac{- 33}{44} & \frac{11}{44} & \frac{1}{44} & \frac{0}{44} & \frac{0}{44} & \frac{0}{44} & \frac{0}{44} \\ 20 & 169 & - 164 & 18 & 105 & 0 & 1 & 0 & 0 & 0 \\ 113 & - 82 & 239 & 218 & - 55 & 0 & 0 & 1 & 0 & 0 \\ 119 & 83 & 7 & 82 & 106 & 0 & 0 & 0 & 1 & 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 20 & 169 & - 164 & 18 & 105 & 0 & 1 & 0 & 0 & 0 \\ 113 & - 82 & 239 & 218 & - 55 & 0 & 0 & 1 & 0 & 0 \\ 119 & 83 & 7 & 82 & 106 & 0 & 0 & 0 & 1 & 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5.2

Perform the row operation $R_{2} = R_{2} - 20 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 5.2.1

Perform the row operation $R_{2} = R_{2} - 20 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 20 - 20 \cdot 1 & 169 - 20 (\frac{75}{44}) & - 164 - 20 (- \frac{15}{11}) & 18 - 20 (- \frac{3}{4}) & 105 - 20 (\frac{1}{4}) & 0 - 20 (\frac{1}{44}) & 1 - 20 \cdot 0 & 0 - 20 \cdot 0 & 0 - 20 \cdot 0 & 0 - 20 \cdot 0 \\ 113 & - 82 & 239 & 218 & - 55 & 0 & 0 & 1 & 0 & 0 \\ 119 & 83 & 7 & 82 & 106 & 0 & 0 & 0 & 1 & 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

Step 5.3

Perform the row operation $R_{3} = R_{3} - 113 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 5.3.1

Perform the row operation $R_{3} = R_{3} - 113 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & \frac{1484}{11} & - \frac{1504}{11} & 33 & 100 & - \frac{5}{11} & 1 & 0 & 0 & 0 \\ 113 - 113 \cdot 1 & - 82 - 113 (\frac{75}{44}) & 239 - 113 (- \frac{15}{11}) & 218 - 113 (- \frac{3}{4}) & - 55 - 113 (\frac{1}{4}) & 0 - 113 (\frac{1}{44}) & 0 - 113 \cdot 0 & 1 - 113 \cdot 0 & 0 - 113 \cdot 0 & 0 - 113 \cdot 0 \\ 119 & 83 & 7 & 82 & 106 & 0 & 0 & 0 & 1 & 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5.3.2

Simplify $R_{3}$ .

Step 5.4

Perform the row operation $R_{4} = R_{4} - 119 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 5.4.1

Perform the row operation $R_{4} = R_{4} - 119 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & \frac{1484}{11} & - \frac{1504}{11} & 33 & 100 & - \frac{5}{11} & 1 & 0 & 0 & 0 \\ 0 & - \frac{12083}{44} & \frac{4324}{11} & \frac{1211}{4} & - \frac{333}{4} & - \frac{113}{44} & 0 & 1 & 0 & 0 \\ 119 - 119 \cdot 1 & 83 - 119 (\frac{75}{44}) & 7 - 119 (- \frac{15}{11}) & 82 - 119 (- \frac{3}{4}) & 106 - 119 (\frac{1}{4}) & 0 - 119 (\frac{1}{44}) & 0 - 119 \cdot 0 & 0 - 119 \cdot 0 & 1 - 119 \cdot 0 & 0 - 119 \cdot 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & \frac{1484}{11} & - \frac{1504}{11} & 33 & 100 & - \frac{5}{11} & 1 & 0 & 0 & 0 \\ 0 & - \frac{12083}{44} & \frac{4324}{11} & \frac{1211}{4} & - \frac{333}{4} & - \frac{113}{44} & 0 & 1 & 0 & 0 \\ 0 & - \frac{5273}{44} & \frac{1862}{11} & \frac{685}{4} & \frac{305}{4} & - \frac{119}{44} & 0 & 0 & 1 & 0 \\ 162 & 20 & 175 & 143 & 74 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Step 5.5

Perform the row operation $R_{5} = R_{5} - 162 R_{1}$ to make the entry at $5, 1$ a $0$ .

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Step 5.5.1

Perform the row operation $R_{5} = R_{5} - 162 R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & \frac{1484}{11} & - \frac{1504}{11} & 33 & 100 & - \frac{5}{11} & 1 & 0 & 0 & 0 \\ 0 & - \frac{12083}{44} & \frac{4324}{11} & \frac{1211}{4} & - \frac{333}{4} & - \frac{113}{44} & 0 & 1 & 0 & 0 \\ 0 & - \frac{5273}{44} & \frac{1862}{11} & \frac{685}{4} & \frac{305}{4} & - \frac{119}{44} & 0 & 0 & 1 & 0 \\ 162 - 162 \cdot 1 & 20 - 162 (\frac{75}{44}) & 175 - 162 (- \frac{15}{11}) & 143 - 162 (- \frac{3}{4}) & 74 - 162 (\frac{1}{4}) & 0 - 162 (\frac{1}{44}) & 0 - 162 \cdot 0 & 0 - 162 \cdot 0 & 0 - 162 \cdot 0 & 1 - 162 \cdot 0 \end{array}]$

Step 5.5.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & \frac{1484}{11} & - \frac{1504}{11} & 33 & 100 & - \frac{5}{11} & 1 & 0 & 0 & 0 \\ 0 & - \frac{12083}{44} & \frac{4324}{11} & \frac{1211}{4} & - \frac{333}{4} & - \frac{113}{44} & 0 & 1 & 0 & 0 \\ 0 & - \frac{5273}{44} & \frac{1862}{11} & \frac{685}{4} & \frac{305}{4} & - \frac{119}{44} & 0 & 0 & 1 & 0 \\ 0 & - \frac{5635}{22} & \frac{4355}{11} & \frac{529}{2} & \frac{67}{2} & - \frac{81}{22} & 0 & 0 & 0 & 1 \end{array}]$

Step 5.6

Multiply each element of $R_{2}$ by $\frac{11}{1484}$ to make the entry at $2, 2$ a $1$ .

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Step 5.6.1

Multiply each element of $R_{2}$ by $\frac{11}{1484}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ \frac{11}{1484} \cdot 0 & \frac{11}{1484} \cdot \frac{1484}{11} & \frac{11}{1484} (- \frac{1504}{11}) & \frac{11}{1484} \cdot 33 & \frac{11}{1484} \cdot 100 & \frac{11}{1484} (- \frac{5}{11}) & \frac{11}{1484} \cdot 1 & \frac{11}{1484} \cdot 0 & \frac{11}{1484} \cdot 0 & \frac{11}{1484} \cdot 0 \\ 0 & - \frac{12083}{44} & \frac{4324}{11} & \frac{1211}{4} & - \frac{333}{4} & - \frac{113}{44} & 0 & 1 & 0 & 0 \\ 0 & - \frac{5273}{44} & \frac{1862}{11} & \frac{685}{4} & \frac{305}{4} & - \frac{119}{44} & 0 & 0 & 1 & 0 \\ 0 & - \frac{5635}{22} & \frac{4355}{11} & \frac{529}{2} & \frac{67}{2} & - \frac{81}{22} & 0 & 0 & 0 & 1 \end{array}]$

Step 5.6.2

Simplify $R_{2}$ .

Step 5.7

Perform the row operation $R_{3} = R_{3} + \frac{12083}{44} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.7.1

Perform the row operation $R_{3} = R_{3} + \frac{12083}{44} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 + \frac{12083}{44} \cdot 0 & - \frac{12083}{44} + \frac{12083}{44} \cdot 1 & \frac{4324}{11} + \frac{12083}{44} (- \frac{376}{371}) & \frac{1211}{4} + \frac{12083}{44} \cdot \frac{363}{1484} & - \frac{333}{4} + \frac{12083}{44} \cdot \frac{275}{371} & - \frac{113}{44} + \frac{12083}{44} (- \frac{5}{1484}) & 0 + \frac{12083}{44} \cdot \frac{11}{1484} & 1 + \frac{12083}{44} \cdot 0 & 0 + \frac{12083}{44} \cdot 0 & 0 + \frac{12083}{44} \cdot 0 \\ 0 & - \frac{5273}{44} & \frac{1862}{11} & \frac{685}{4} & \frac{305}{4} & - \frac{119}{44} & 0 & 0 & 1 & 0 \\ 0 & - \frac{5635}{22} & \frac{4355}{11} & \frac{529}{2} & \frac{67}{2} & - \frac{81}{22} & 0 & 0 & 0 & 1 \end{array}]$

Step 5.7.2

Simplify $R_{3}$ .

Step 5.8

Perform the row operation $R_{4} = R_{4} + \frac{5273}{44} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 5.8.1

Perform the row operation $R_{4} = R_{4} + \frac{5273}{44} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & \frac{42582}{371} & \frac{2195863}{5936} & \frac{44633}{371} & - \frac{20737}{5936} & \frac{12083}{5936} & 1 & 0 & 0 \\ 0 + \frac{5273}{44} \cdot 0 & - \frac{5273}{44} + \frac{5273}{44} \cdot 1 & \frac{1862}{11} + \frac{5273}{44} (- \frac{376}{371}) & \frac{685}{4} + \frac{5273}{44} \cdot \frac{363}{1484} & \frac{305}{4} + \frac{5273}{44} \cdot \frac{275}{371} & - \frac{119}{44} + \frac{5273}{44} (- \frac{5}{1484}) & 0 + \frac{5273}{44} \cdot \frac{11}{1484} & 0 + \frac{5273}{44} \cdot 0 & 1 + \frac{5273}{44} \cdot 0 & 0 + \frac{5273}{44} \cdot 0 \\ 0 & - \frac{5635}{22} & \frac{4355}{11} & \frac{529}{2} & \frac{67}{2} & - \frac{81}{22} & 0 & 0 & 0 & 1 \end{array}]$

Step 5.8.2

Simplify $R_{4}$ .

Step 5.9

Perform the row operation $R_{5} = R_{5} + \frac{5635}{22} R_{2}$ to make the entry at $5, 2$ a $0$ .

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Step 5.9.1

Perform the row operation $R_{5} = R_{5} + \frac{5635}{22} R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & \frac{42582}{371} & \frac{2195863}{5936} & \frac{44633}{371} & - \frac{20737}{5936} & \frac{12083}{5936} & 1 & 0 & 0 \\ 0 & 0 & \frac{17740}{371} & \frac{1190549}{5936} & \frac{61245}{371} & - \frac{18451}{5936} & \frac{5273}{5936} & 0 & 1 & 0 \\ 0 + \frac{5635}{22} \cdot 0 & - \frac{5635}{22} + \frac{5635}{22} \cdot 1 & \frac{4355}{11} + \frac{5635}{22} (- \frac{376}{371}) & \frac{529}{2} + \frac{5635}{22} \cdot \frac{363}{1484} & \frac{67}{2} + \frac{5635}{22} \cdot \frac{275}{371} & - \frac{81}{22} + \frac{5635}{22} (- \frac{5}{1484}) & 0 + \frac{5635}{22} \cdot \frac{11}{1484} & 0 + \frac{5635}{22} \cdot 0 & 0 + \frac{5635}{22} \cdot 0 & 1 + \frac{5635}{22} \cdot 0 \end{array}]$

Step 5.9.2

Simplify $R_{5}$ .

Step 5.10

Multiply each element of $R_{3}$ by $\frac{371}{42582}$ to make the entry at $3, 3$ a $1$ .

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Step 5.10.1

Multiply each element of $R_{3}$ by $\frac{371}{42582}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ \frac{371}{42582} \cdot 0 & \frac{371}{42582} \cdot 0 & \frac{371}{42582} \cdot \frac{42582}{371} & \frac{371}{42582} \cdot \frac{2195863}{5936} & \frac{371}{42582} \cdot \frac{44633}{371} & \frac{371}{42582} (- \frac{20737}{5936}) & \frac{371}{42582} \cdot \frac{12083}{5936} & \frac{371}{42582} \cdot 1 & \frac{371}{42582} \cdot 0 & \frac{371}{42582} \cdot 0 \\ 0 & 0 & \frac{17740}{371} & \frac{1190549}{5936} & \frac{61245}{371} & - \frac{18451}{5936} & \frac{5273}{5936} & 0 & 1 & 0 \\ 0 & 0 & \frac{7225}{53} & \frac{138713}{424} & \frac{11838}{53} & - \frac{1927}{424} & \frac{805}{424} & 0 & 0 & 1 \end{array}]$

Step 5.10.2

Simplify $R_{3}$ .

Step 5.11

Perform the row operation $R_{4} = R_{4} - \frac{17740}{371} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 5.11.1

Perform the row operation $R_{4} = R_{4} - \frac{17740}{371} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & \frac{44633}{42582} & - \frac{20737}{681312} & \frac{12083}{681312} & \frac{371}{42582} & 0 & 0 \\ 0 - \frac{17740}{371} \cdot 0 & 0 - \frac{17740}{371} \cdot 0 & \frac{17740}{371} - \frac{17740}{371} \cdot 1 & \frac{1190549}{5936} - \frac{17740}{371} \cdot \frac{2195863}{681312} & \frac{61245}{371} - \frac{17740}{371} \cdot \frac{44633}{42582} & - \frac{18451}{5936} - \frac{17740}{371} (- \frac{20737}{681312}) & \frac{5273}{5936} - \frac{17740}{371} \cdot \frac{12083}{681312} & 0 - \frac{17740}{371} \cdot \frac{371}{42582} & 1 - \frac{17740}{371} \cdot 0 & 0 - \frac{17740}{371} \cdot 0 \\ 0 & 0 & \frac{7225}{53} & \frac{138713}{424} & \frac{11838}{53} & - \frac{1927}{424} & \frac{805}{424} & 0 & 0 & 1 \end{array}]$

Step 5.11.2

Simplify $R_{4}$ .

Step 5.12

Perform the row operation $R_{5} = R_{5} - \frac{7225}{53} R_{3}$ to make the entry at $5, 3$ a $0$ .

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Step 5.12.1

Perform the row operation $R_{5} = R_{5} - \frac{7225}{53} R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & \frac{44633}{42582} & - \frac{20737}{681312} & \frac{12083}{681312} & \frac{371}{42582} & 0 & 0 \\ 0 & 0 & 0 & \frac{15823919}{340656} & \frac{2447635}{21291} & - \frac{563081}{340656} & \frac{13723}{340656} & - \frac{8870}{21291} & 1 & 0 \\ 0 - \frac{7225}{53} \cdot 0 & 0 - \frac{7225}{53} \cdot 0 & \frac{7225}{53} - \frac{7225}{53} \cdot 1 & \frac{138713}{424} - \frac{7225}{53} \cdot \frac{2195863}{681312} & \frac{11838}{53} - \frac{7225}{53} \cdot \frac{44633}{42582} & - \frac{1927}{424} - \frac{7225}{53} (- \frac{20737}{681312}) & \frac{805}{424} - \frac{7225}{53} \cdot \frac{12083}{681312} & 0 - \frac{7225}{53} \cdot \frac{371}{42582} & 0 - \frac{7225}{53} \cdot 0 & 1 - \frac{7225}{53} \cdot 0 \end{array}]$

Step 5.12.2

Simplify $R_{5}$ .

Step 5.13

Multiply each element of $R_{4}$ by $\frac{340656}{15823919}$ to make the entry at $4, 4$ a $1$ .

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Step 5.13.1

Multiply each element of $R_{4}$ by $\frac{340656}{15823919}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & \frac{44633}{42582} & - \frac{20737}{681312} & \frac{12083}{681312} & \frac{371}{42582} & 0 & 0 \\ \frac{340656}{15823919} \cdot 0 & \frac{340656}{15823919} \cdot 0 & \frac{340656}{15823919} \cdot 0 & \frac{340656}{15823919} \cdot \frac{15823919}{340656} & \frac{340656}{15823919} \cdot \frac{2447635}{21291} & \frac{340656}{15823919} (- \frac{563081}{340656}) & \frac{340656}{15823919} \cdot \frac{13723}{340656} & \frac{340656}{15823919} (- \frac{8870}{21291}) & \frac{340656}{15823919} \cdot 1 & \frac{340656}{15823919} \cdot 0 \\ 0 & 0 & 0 & - \frac{76448231}{681312} & \frac{3426647}{42582} & - \frac{269551}{681312} & - \frac{353635}{681312} & - \frac{50575}{42582} & 0 & 1 \end{array}]$

Step 5.13.2

Simplify $R_{4}$ .

Step 5.14

Perform the row operation $R_{5} = R_{5} + \frac{76448231}{681312} R_{4}$ to make the entry at $5, 4$ a $0$ .

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Step 5.14.1

Perform the row operation $R_{5} = R_{5} + \frac{76448231}{681312} R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & \frac{44633}{42582} & - \frac{20737}{681312} & \frac{12083}{681312} & \frac{371}{42582} & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{39162160}{15823919} & - \frac{563081}{15823919} & \frac{13723}{15823919} & - \frac{141920}{15823919} & \frac{340656}{15823919} & 0 \\ 0 + \frac{76448231}{681312} \cdot 0 & 0 + \frac{76448231}{681312} \cdot 0 & 0 + \frac{76448231}{681312} \cdot 0 & - \frac{76448231}{681312} + \frac{76448231}{681312} \cdot 1 & \frac{3426647}{42582} + \frac{76448231}{681312} \cdot \frac{39162160}{15823919} & - \frac{269551}{681312} + \frac{76448231}{681312} (- \frac{563081}{15823919}) & - \frac{353635}{681312} + \frac{76448231}{681312} \cdot \frac{13723}{15823919} & - \frac{50575}{42582} + \frac{76448231}{681312} (- \frac{141920}{15823919}) & 0 + \frac{76448231}{681312} \cdot \frac{340656}{15823919} & 1 + \frac{76448231}{681312} \cdot 0 \end{array}]$

Step 5.14.2

Simplify $R_{5}$ .

Step 5.15

Multiply each element of $R_{5}$ by $\frac{15823919}{5667661229}$ to make the entry at $5, 5$ a $1$ .

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Step 5.15.1

Multiply each element of $R_{5}$ by $\frac{15823919}{5667661229}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & \frac{44633}{42582} & - \frac{20737}{681312} & \frac{12083}{681312} & \frac{371}{42582} & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{39162160}{15823919} & - \frac{563081}{15823919} & \frac{13723}{15823919} & - \frac{141920}{15823919} & \frac{340656}{15823919} & 0 \\ \frac{15823919}{5667661229} \cdot 0 & \frac{15823919}{5667661229} \cdot 0 & \frac{15823919}{5667661229} \cdot 0 & \frac{15823919}{5667661229} \cdot 0 & \frac{15823919}{5667661229} \cdot \frac{5667661229}{15823919} & \frac{15823919}{5667661229} (- \frac{69442340}{15823919}) & \frac{15823919}{5667661229} (- \frac{13347167}{31647838}) & \frac{15823919}{5667661229} (- \frac{69437345}{31647838}) & \frac{15823919}{5667661229} \cdot \frac{76448231}{31647838} & \frac{15823919}{5667661229} \cdot 1 \end{array}]$

Step 5.15.2

Simplify $R_{5}$ .

Step 5.16

Perform the row operation $R_{4} = R_{4} - \frac{39162160}{15823919} R_{5}$ to make the entry at $4, 5$ a $0$ .

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Step 5.16.1

Perform the row operation $R_{4} = R_{4} - \frac{39162160}{15823919} R_{5}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & \frac{44633}{42582} & - \frac{20737}{681312} & \frac{12083}{681312} & \frac{371}{42582} & 0 & 0 \\ 0 - \frac{39162160}{15823919} \cdot 0 & 0 - \frac{39162160}{15823919} \cdot 0 & 0 - \frac{39162160}{15823919} \cdot 0 & 1 - \frac{39162160}{15823919} \cdot 0 & \frac{39162160}{15823919} - \frac{39162160}{15823919} \cdot 1 & - \frac{563081}{15823919} - \frac{39162160}{15823919} (- \frac{69442340}{5667661229}) & \frac{13723}{15823919} - \frac{39162160}{15823919} (- \frac{13347167}{11335322458}) & - \frac{141920}{15823919} - \frac{39162160}{15823919} (- \frac{69437345}{11335322458}) & \frac{340656}{15823919} - \frac{39162160}{15823919} \cdot \frac{76448231}{11335322458} & 0 - \frac{39162160}{15823919} \cdot \frac{15823919}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.16.2

Simplify $R_{4}$ .

Step 5.17

Perform the row operation $R_{3} = R_{3} - \frac{44633}{42582} R_{5}$ to make the entry at $3, 5$ a $0$ .

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Step 5.17.1

Perform the row operation $R_{3} = R_{3} - \frac{44633}{42582} R_{5}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 - \frac{44633}{42582} \cdot 0 & 0 - \frac{44633}{42582} \cdot 0 & 1 - \frac{44633}{42582} \cdot 0 & \frac{2195863}{681312} - \frac{44633}{42582} \cdot 0 & \frac{44633}{42582} - \frac{44633}{42582} \cdot 1 & - \frac{20737}{681312} - \frac{44633}{42582} (- \frac{69442340}{5667661229}) & \frac{12083}{681312} - \frac{44633}{42582} (- \frac{13347167}{11335322458}) & \frac{371}{42582} - \frac{44633}{42582} (- \frac{69437345}{11335322458}) & 0 - \frac{44633}{42582} \cdot \frac{76448231}{11335322458} & 0 - \frac{44633}{42582} \cdot \frac{15823919}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.17.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & \frac{275}{371} & - \frac{5}{1484} & \frac{11}{1484} & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{2195863}{681312} & 0 & - \frac{67939571526253}{3861445607252448} & \frac{24416047822565}{1287148535750816} & \frac{2434867217101}{160893566968852} & - \frac{3412113894223}{482680700906556} & - \frac{706268976727}{241340350453278} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.18

Perform the row operation $R_{2} = R_{2} - \frac{275}{371} R_{5}$ to make the entry at $2, 5$ a $0$ .

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Step 5.18.1

Perform the row operation $R_{2} = R_{2} - \frac{275}{371} R_{5}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 - \frac{275}{371} \cdot 0 & 1 - \frac{275}{371} \cdot 0 & - \frac{376}{371} - \frac{275}{371} \cdot 0 & \frac{363}{1484} - \frac{275}{371} \cdot 0 & \frac{275}{371} - \frac{275}{371} \cdot 1 & - \frac{5}{1484} - \frac{275}{371} (- \frac{69442340}{5667661229}) & \frac{11}{1484} - \frac{275}{371} (- \frac{13347167}{11335322458}) & 0 - \frac{275}{371} (- \frac{69437345}{11335322458}) & 0 - \frac{275}{371} \cdot \frac{76448231}{11335322458} & 0 - \frac{275}{371} \cdot \frac{15823919}{5667661229} \\ 0 & 0 & 1 & \frac{2195863}{681312} & 0 & - \frac{67939571526253}{3861445607252448} & \frac{24416047822565}{1287148535750816} & \frac{2434867217101}{160893566968852} & - \frac{3412113894223}{482680700906556} & - \frac{706268976727}{241340350453278} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.18.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & \frac{1}{4} & \frac{1}{44} & 0 & 0 & 0 & 0 \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & 0 & \frac{6864038265}{1201544180548} & \frac{9955030767}{1201544180548} & \frac{19095269875}{4205404631918} & - \frac{21023263525}{4205404631918} & - \frac{4351577725}{2102702315959} \\ 0 & 0 & 1 & \frac{2195863}{681312} & 0 & - \frac{67939571526253}{3861445607252448} & \frac{24416047822565}{1287148535750816} & \frac{2434867217101}{160893566968852} & - \frac{3412113894223}{482680700906556} & - \frac{706268976727}{241340350453278} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.19

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{5}$ to make the entry at $1, 5$ a $0$ .

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Step 5.19.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{5}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{4} \cdot 0 & \frac{75}{44} - \frac{1}{4} \cdot 0 & - \frac{15}{11} - \frac{1}{4} \cdot 0 & - \frac{3}{4} - \frac{1}{4} \cdot 0 & \frac{1}{4} - \frac{1}{4} \cdot 1 & \frac{1}{44} - \frac{1}{4} (- \frac{69442340}{5667661229}) & 0 - \frac{1}{4} (- \frac{13347167}{11335322458}) & 0 - \frac{1}{4} (- \frac{69437345}{11335322458}) & 0 - \frac{1}{4} \cdot \frac{76448231}{11335322458} & 0 - \frac{1}{4} \cdot \frac{15823919}{5667661229} \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & 0 & \frac{6864038265}{1201544180548} & \frac{9955030767}{1201544180548} & \frac{19095269875}{4205404631918} & - \frac{21023263525}{4205404631918} & - \frac{4351577725}{2102702315959} \\ 0 & 0 & 1 & \frac{2195863}{681312} & 0 & - \frac{67939571526253}{3861445607252448} & \frac{24416047822565}{1287148535750816} & \frac{2434867217101}{160893566968852} & - \frac{3412113894223}{482680700906556} & - \frac{706268976727}{241340350453278} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.19.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & 0 & \frac{6431526969}{249377094076} & \frac{13347167}{45341289832} & \frac{69437345}{45341289832} & - \frac{76448231}{45341289832} & - \frac{15823919}{22670644916} \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & 0 & \frac{6864038265}{1201544180548} & \frac{9955030767}{1201544180548} & \frac{19095269875}{4205404631918} & - \frac{21023263525}{4205404631918} & - \frac{4351577725}{2102702315959} \\ 0 & 0 & 1 & \frac{2195863}{681312} & 0 & - \frac{67939571526253}{3861445607252448} & \frac{24416047822565}{1287148535750816} & \frac{2434867217101}{160893566968852} & - \frac{3412113894223}{482680700906556} & - \frac{706268976727}{241340350453278} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.20

Perform the row operation $R_{3} = R_{3} - \frac{2195863}{681312} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 5.20.1

Perform the row operation $R_{3} = R_{3} - \frac{2195863}{681312} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & 0 & \frac{6431526969}{249377094076} & \frac{13347167}{45341289832} & \frac{69437345}{45341289832} & - \frac{76448231}{45341289832} & - \frac{15823919}{22670644916} \\ 0 & 1 & - \frac{376}{371} & \frac{363}{1484} & 0 & \frac{6864038265}{1201544180548} & \frac{9955030767}{1201544180548} & \frac{19095269875}{4205404631918} & - \frac{21023263525}{4205404631918} & - \frac{4351577725}{2102702315959} \\ 0 - \frac{2195863}{681312} \cdot 0 & 0 - \frac{2195863}{681312} \cdot 0 & 1 - \frac{2195863}{681312} \cdot 0 & \frac{2195863}{681312} - \frac{2195863}{681312} \cdot 1 & 0 - \frac{2195863}{681312} \cdot 0 & - \frac{67939571526253}{3861445607252448} - \frac{2195863}{681312} (- \frac{29818171}{5667661229}) & \frac{24416047822565}{1287148535750816} - \frac{2195863}{681312} \cdot \frac{21431433}{5667661229} & \frac{2434867217101}{160893566968852} - \frac{2195863}{681312} \cdot \frac{35092680}{5667661229} & - \frac{3412113894223}{482680700906556} - \frac{2195863}{681312} \cdot \frac{27413176}{5667661229} & - \frac{706268976727}{241340350453278} - \frac{2195863}{681312} (- \frac{39162160}{5667661229}) \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.20.2

Simplify $R_{3}$ .

Step 5.21

Perform the row operation $R_{2} = R_{2} - \frac{363}{1484} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 5.21.1

Perform the row operation $R_{2} = R_{2} - \frac{363}{1484} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & - \frac{3}{4} & 0 & \frac{6431526969}{249377094076} & \frac{13347167}{45341289832} & \frac{69437345}{45341289832} & - \frac{76448231}{45341289832} & - \frac{15823919}{22670644916} \\ 0 - \frac{363}{1484} \cdot 0 & 1 - \frac{363}{1484} \cdot 0 & - \frac{376}{371} - \frac{363}{1484} \cdot 0 & \frac{363}{1484} - \frac{363}{1484} \cdot 1 & 0 - \frac{363}{1484} \cdot 0 & \frac{6864038265}{1201544180548} - \frac{363}{1484} (- \frac{29818171}{5667661229}) & \frac{9955030767}{1201544180548} - \frac{363}{1484} \cdot \frac{21431433}{5667661229} & \frac{19095269875}{4205404631918} - \frac{363}{1484} \cdot \frac{35092680}{5667661229} & - \frac{21023263525}{4205404631918} - \frac{363}{1484} \cdot \frac{27413176}{5667661229} & - \frac{4351577725}{2102702315959} - \frac{363}{1484} (- \frac{39162160}{5667661229}) \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.21.2

Simplify $R_{2}$ .

Step 5.22

Perform the row operation $R_{1} = R_{1} + \frac{3}{4} R_{4}$ to make the entry at $1, 4$ a $0$ .

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Step 5.22.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{4} R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{4} \cdot 0 & \frac{75}{44} + \frac{3}{4} \cdot 0 & - \frac{15}{11} + \frac{3}{4} \cdot 0 & - \frac{3}{4} + \frac{3}{4} \cdot 1 & 0 + \frac{3}{4} \cdot 0 & \frac{6431526969}{249377094076} + \frac{3}{4} (- \frac{29818171}{5667661229}) & \frac{13347167}{45341289832} + \frac{3}{4} \cdot \frac{21431433}{5667661229} & \frac{69437345}{45341289832} + \frac{3}{4} \cdot \frac{35092680}{5667661229} & - \frac{76448231}{45341289832} + \frac{3}{4} \cdot \frac{27413176}{5667661229} & - \frac{15823919}{22670644916} + \frac{3}{4} (- \frac{39162160}{5667661229}) \\ 0 & 1 & - \frac{376}{371} & 0 & 0 & \frac{14718065982}{2102702315959} & \frac{30952802595}{4205404631918} & \frac{240112235}{79347257206} & - \frac{25998754969}{4205404631918} & - \frac{797611705}{2102702315959} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.22.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & 0 & 0 & \frac{1892817}{86649442} & \frac{98635}{31508888} & \frac{194575}{31508888} & \frac{61175}{31508888} & - \frac{92641}{15754444} \\ 0 & 1 & - \frac{376}{371} & 0 & 0 & \frac{14718065982}{2102702315959} & \frac{30952802595}{4205404631918} & \frac{240112235}{79347257206} & - \frac{25998754969}{4205404631918} & - \frac{797611705}{2102702315959} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.23

Perform the row operation $R_{2} = R_{2} + \frac{376}{371} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 5.23.1

Perform the row operation $R_{2} = R_{2} + \frac{376}{371} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & 0 & 0 & \frac{1892817}{86649442} & \frac{98635}{31508888} & \frac{194575}{31508888} & \frac{61175}{31508888} & - \frac{92641}{15754444} \\ 0 + \frac{376}{371} \cdot 0 & 1 + \frac{376}{371} \cdot 0 & - \frac{376}{371} + \frac{376}{371} \cdot 1 & 0 + \frac{376}{371} \cdot 0 & 0 + \frac{376}{371} \cdot 0 & \frac{14718065982}{2102702315959} + \frac{376}{371} (- \frac{3615015}{5667661229}) & \frac{30952802595}{4205404631918} + \frac{376}{371} \cdot \frac{38437093}{5667661229} & \frac{240112235}{79347257206} + \frac{376}{371} (- \frac{27332418}{5667661229}) & - \frac{25998754969}{4205404631918} + \frac{376}{371} (- \frac{128417656}{5667661229}) & - \frac{797611705}{2102702315959} + \frac{376}{371} \cdot \frac{109633229}{5667661229} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.23.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & - \frac{15}{11} & 0 & 0 & \frac{1892817}{86649442} & \frac{98635}{31508888} & \frac{194575}{31508888} & \frac{61175}{31508888} & - \frac{92641}{15754444} \\ 0 & 1 & 0 & 0 & 0 & \frac{36007602}{5667661229} & \frac{161340961}{11335322458} & - \frac{21099811}{11335322458} & - \frac{330374211}{11335322458} & \frac{108960869}{5667661229} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.24

Perform the row operation $R_{1} = R_{1} + \frac{15}{11} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 5.24.1

Perform the row operation $R_{1} = R_{1} + \frac{15}{11} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + \frac{15}{11} \cdot 0 & \frac{75}{44} + \frac{15}{11} \cdot 0 & - \frac{15}{11} + \frac{15}{11} \cdot 1 & 0 + \frac{15}{11} \cdot 0 & 0 + \frac{15}{11} \cdot 0 & \frac{1892817}{86649442} + \frac{15}{11} (- \frac{3615015}{5667661229}) & \frac{98635}{31508888} + \frac{15}{11} \cdot \frac{38437093}{5667661229} & \frac{194575}{31508888} + \frac{15}{11} (- \frac{27332418}{5667661229}) & \frac{61175}{31508888} + \frac{15}{11} (- \frac{128417656}{5667661229}) & - \frac{92641}{15754444} + \frac{15}{11} \cdot \frac{109633229}{5667661229} \\ 0 & 1 & 0 & 0 & 0 & \frac{36007602}{5667661229} & \frac{161340961}{11335322458} & - \frac{21099811}{11335322458} & - \frac{330374211}{11335322458} & \frac{108960869}{5667661229} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.24.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{75}{44} & 0 & 0 & 0 & \frac{2615313213}{124688547038} & \frac{6173744575}{498754188152} & - \frac{199962485}{498754188152} & - \frac{14441779645}{498754188152} & \frac{5111579351}{249377094076} \\ 0 & 1 & 0 & 0 & 0 & \frac{36007602}{5667661229} & \frac{161340961}{11335322458} & - \frac{21099811}{11335322458} & - \frac{330374211}{11335322458} & \frac{108960869}{5667661229} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.25

Perform the row operation $R_{1} = R_{1} - \frac{75}{44} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 5.25.1

Perform the row operation $R_{1} = R_{1} - \frac{75}{44} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{75}{44} \cdot 0 & \frac{75}{44} - \frac{75}{44} \cdot 1 & 0 - \frac{75}{44} \cdot 0 & 0 - \frac{75}{44} \cdot 0 & 0 - \frac{75}{44} \cdot 0 & \frac{2615313213}{124688547038} - \frac{75}{44} \cdot \frac{36007602}{5667661229} & \frac{6173744575}{498754188152} - \frac{75}{44} \cdot \frac{161340961}{11335322458} & - \frac{199962485}{498754188152} - \frac{75}{44} (- \frac{21099811}{11335322458}) & - \frac{14441779645}{498754188152} - \frac{75}{44} (- \frac{330374211}{11335322458}) & \frac{5111579351}{249377094076} - \frac{75}{44} \cdot \frac{108960869}{5667661229} \\ 0 & 1 & 0 & 0 & 0 & \frac{36007602}{5667661229} & \frac{161340961}{11335322458} & - \frac{21099811}{11335322458} & - \frac{330374211}{11335322458} & \frac{108960869}{5667661229} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 5.25.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & \frac{57501279}{5667661229} & - \frac{134700625}{11335322458} & \frac{31420985}{11335322458} & \frac{234915595}{11335322458} & - \frac{69556496}{5667661229} \\ 0 & 1 & 0 & 0 & 0 & \frac{36007602}{5667661229} & \frac{161340961}{11335322458} & - \frac{21099811}{11335322458} & - \frac{330374211}{11335322458} & \frac{108960869}{5667661229} \\ 0 & 0 & 1 & 0 & 0 & - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ 0 & 0 & 0 & 1 & 0 & - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ 0 & 0 & 0 & 0 & 1 & - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$

Step 6

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} \frac{57501279}{5667661229} & - \frac{134700625}{11335322458} & \frac{31420985}{11335322458} & \frac{234915595}{11335322458} & - \frac{69556496}{5667661229} \\ \frac{36007602}{5667661229} & \frac{161340961}{11335322458} & - \frac{21099811}{11335322458} & - \frac{330374211}{11335322458} & \frac{108960869}{5667661229} \\ - \frac{3615015}{5667661229} & \frac{38437093}{5667661229} & - \frac{27332418}{5667661229} & - \frac{128417656}{5667661229} & \frac{109633229}{5667661229} \\ - \frac{29818171}{5667661229} & \frac{21431433}{5667661229} & \frac{35092680}{5667661229} & \frac{27413176}{5667661229} & - \frac{39162160}{5667661229} \\ - \frac{69442340}{5667661229} & - \frac{13347167}{11335322458} & - \frac{69437345}{11335322458} & \frac{76448231}{11335322458} & \frac{15823919}{5667661229} \end{array}]$