Finite Math Examples

$[\begin{array}{c} 12 & 6 & 7 & 1 \\ 21 & 7 & 22 & 3 \\ 28 & 8 & 26 & 21 \\ 37 & 29 & 27 & 24 \end{array}]$

Step 1

Consider the corresponding sign chart.

$[\begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array}]$

Step 2

Use the sign chart and the given matrix to find the cofactor of each element.

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Step 2.1

Calculate the minor for element $a_{11}$ .

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Step 2.1.1

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

$| \begin{array}{c} 7 & 22 & 3 \\ 8 & 26 & 21 \\ 29 & 27 & 24 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

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 2.1.2.1.1

Consider the corresponding sign chart.

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

Step 2.1.2.1.2

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

Step 2.1.2.1.3

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

$| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.1.2.1.4

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

$7 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.1.2.1.5

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

$| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.1.2.1.6

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

$- 22 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.1.2.1.7

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

$| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.1.8

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

$3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.1.9

Add the terms together.

$a_{11} = 7 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} | - 22 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.2

Evaluate $| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$ .

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Step 2.1.2.2.1

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

$a_{11} = 7 (26 \cdot 24 - 27 \cdot 21) - 22 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.2.2

Simplify the determinant.

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Step 2.1.2.2.2.1

Simplify each term.

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Step 2.1.2.2.2.1.1

Multiply $26$ by $24$ .

$a_{11} = 7 (624 - 27 \cdot 21) - 22 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.2.2.1.2

Multiply $- 27$ by $21$ .

$a_{11} = 7 (624 - 567) - 22 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.2.2.2

Subtract $567$ from $624$ .

$a_{11} = 7 \cdot 57 - 22 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.3

Evaluate $| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$ .

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Step 2.1.2.3.1

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

$a_{11} = 7 \cdot 57 - 22 (8 \cdot 24 - 29 \cdot 21) + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.3.2

Simplify the determinant.

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Step 2.1.2.3.2.1

Simplify each term.

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Step 2.1.2.3.2.1.1

Multiply $8$ by $24$ .

$a_{11} = 7 \cdot 57 - 22 (192 - 29 \cdot 21) + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.3.2.1.2

Multiply $- 29$ by $21$ .

$a_{11} = 7 \cdot 57 - 22 (192 - 609) + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.3.2.2

Subtract $609$ from $192$ .

$a_{11} = 7 \cdot 57 - 22 \cdot - 417 + 3 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.1.2.4

Evaluate $| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$ .

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Step 2.1.2.4.1

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

$a_{11} = 7 \cdot 57 - 22 \cdot - 417 + 3 (8 \cdot 27 - 29 \cdot 26)$

Step 2.1.2.4.2

Simplify the determinant.

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Step 2.1.2.4.2.1

Simplify each term.

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Step 2.1.2.4.2.1.1

Multiply $8$ by $27$ .

$a_{11} = 7 \cdot 57 - 22 \cdot - 417 + 3 (216 - 29 \cdot 26)$

Step 2.1.2.4.2.1.2

Multiply $- 29$ by $26$ .

$a_{11} = 7 \cdot 57 - 22 \cdot - 417 + 3 (216 - 754)$

Step 2.1.2.4.2.2

Subtract $754$ from $216$ .

$a_{11} = 7 \cdot 57 - 22 \cdot - 417 + 3 \cdot - 538$

Step 2.1.2.5

Simplify the determinant.

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Step 2.1.2.5.1

Simplify each term.

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Step 2.1.2.5.1.1

Multiply $7$ by $57$ .

$a_{11} = 399 - 22 \cdot - 417 + 3 \cdot - 538$

Step 2.1.2.5.1.2

Multiply $- 22$ by $- 417$ .

$a_{11} = 399 + 9174 + 3 \cdot - 538$

Step 2.1.2.5.1.3

Multiply $3$ by $- 538$ .

$a_{11} = 399 + 9174 - 1614$

Step 2.1.2.5.2

Add $399$ and $9174$ .

$a_{11} = 9573 - 1614$

Step 2.1.2.5.3

Subtract $1614$ from $9573$ .

$a_{11} = 7959$

Step 2.2

Calculate the minor for element $a_{12}$ .

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Step 2.2.1

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

$| \begin{array}{c} 21 & 22 & 3 \\ 28 & 26 & 21 \\ 37 & 27 & 24 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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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.

$| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.2.2.1.4

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

$21 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.2.2.1.5

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

$| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.2.2.1.6

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

$- 22 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.2.2.1.7

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

$| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.1.8

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

$3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.1.9

Add the terms together.

$a_{12} = 21 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} | - 22 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.2

Evaluate $| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$ .

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Step 2.2.2.2.1

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

$a_{12} = 21 (26 \cdot 24 - 27 \cdot 21) - 22 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

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 $26$ by $24$ .

$a_{12} = 21 (624 - 27 \cdot 21) - 22 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.2.2.1.2

Multiply $- 27$ by $21$ .

$a_{12} = 21 (624 - 567) - 22 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.2.2.2

Subtract $567$ from $624$ .

$a_{12} = 21 \cdot 57 - 22 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.3

Evaluate $| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$ .

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Step 2.2.2.3.1

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

$a_{12} = 21 \cdot 57 - 22 (28 \cdot 24 - 37 \cdot 21) + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

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 $28$ by $24$ .

$a_{12} = 21 \cdot 57 - 22 (672 - 37 \cdot 21) + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.3.2.1.2

Multiply $- 37$ by $21$ .

$a_{12} = 21 \cdot 57 - 22 (672 - 777) + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.3.2.2

Subtract $777$ from $672$ .

$a_{12} = 21 \cdot 57 - 22 \cdot - 105 + 3 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.2.2.4

Evaluate $| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$ .

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Step 2.2.2.4.1

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

$a_{12} = 21 \cdot 57 - 22 \cdot - 105 + 3 (28 \cdot 27 - 37 \cdot 26)$

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 $28$ by $27$ .

$a_{12} = 21 \cdot 57 - 22 \cdot - 105 + 3 (756 - 37 \cdot 26)$

Step 2.2.2.4.2.1.2

Multiply $- 37$ by $26$ .

$a_{12} = 21 \cdot 57 - 22 \cdot - 105 + 3 (756 - 962)$

Step 2.2.2.4.2.2

Subtract $962$ from $756$ .

$a_{12} = 21 \cdot 57 - 22 \cdot - 105 + 3 \cdot - 206$

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 $21$ by $57$ .

$a_{12} = 1197 - 22 \cdot - 105 + 3 \cdot - 206$

Step 2.2.2.5.1.2

Multiply $- 22$ by $- 105$ .

$a_{12} = 1197 + 2310 + 3 \cdot - 206$

Step 2.2.2.5.1.3

Multiply $3$ by $- 206$ .

$a_{12} = 1197 + 2310 - 618$

Step 2.2.2.5.2

Add $1197$ and $2310$ .

$a_{12} = 3507 - 618$

Step 2.2.2.5.3

Subtract $618$ from $3507$ .

$a_{12} = 2889$

Step 2.3

Calculate the minor for element $a_{13}$ .

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Step 2.3.1

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

$| \begin{array}{c} 21 & 7 & 3 \\ 28 & 8 & 21 \\ 37 & 29 & 24 \end{array} |$

Step 2.3.2

Evaluate the determinant.

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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.

$| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.3.2.1.4

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

$21 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.3.2.1.5

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

$| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.3.2.1.6

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

$- 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.3.2.1.7

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

$| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.1.8

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

$3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.1.9

Add the terms together.

$a_{13} = 21 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.2

Evaluate $| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$ .

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Step 2.3.2.2.1

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

$a_{13} = 21 (8 \cdot 24 - 29 \cdot 21) - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

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 $8$ by $24$ .

$a_{13} = 21 (192 - 29 \cdot 21) - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.2.2.1.2

Multiply $- 29$ by $21$ .

$a_{13} = 21 (192 - 609) - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.2.2.2

Subtract $609$ from $192$ .

$a_{13} = 21 \cdot - 417 - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.3

Evaluate $| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$ .

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Step 2.3.2.3.1

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

$a_{13} = 21 \cdot - 417 - 7 (28 \cdot 24 - 37 \cdot 21) + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

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 $28$ by $24$ .

$a_{13} = 21 \cdot - 417 - 7 (672 - 37 \cdot 21) + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.3.2.1.2

Multiply $- 37$ by $21$ .

$a_{13} = 21 \cdot - 417 - 7 (672 - 777) + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.3.2.2

Subtract $777$ from $672$ .

$a_{13} = 21 \cdot - 417 - 7 \cdot - 105 + 3 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.3.2.4

Evaluate $| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$ .

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Step 2.3.2.4.1

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

$a_{13} = 21 \cdot - 417 - 7 \cdot - 105 + 3 (28 \cdot 29 - 37 \cdot 8)$

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 $28$ by $29$ .

$a_{13} = 21 \cdot - 417 - 7 \cdot - 105 + 3 (812 - 37 \cdot 8)$

Step 2.3.2.4.2.1.2

Multiply $- 37$ by $8$ .

$a_{13} = 21 \cdot - 417 - 7 \cdot - 105 + 3 (812 - 296)$

Step 2.3.2.4.2.2

Subtract $296$ from $812$ .

$a_{13} = 21 \cdot - 417 - 7 \cdot - 105 + 3 \cdot 516$

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 $21$ by $- 417$ .

$a_{13} = - 8757 - 7 \cdot - 105 + 3 \cdot 516$

Step 2.3.2.5.1.2

Multiply $- 7$ by $- 105$ .

$a_{13} = - 8757 + 735 + 3 \cdot 516$

Step 2.3.2.5.1.3

Multiply $3$ by $516$ .

$a_{13} = - 8757 + 735 + 1548$

Step 2.3.2.5.2

Add $- 8757$ and $735$ .

$a_{13} = - 8022 + 1548$

Step 2.3.2.5.3

Add $- 8022$ and $1548$ .

$a_{13} = - 6474$

Step 2.4

Calculate the minor for element $a_{14}$ .

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Step 2.4.1

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

$| \begin{array}{c} 21 & 7 & 22 \\ 28 & 8 & 26 \\ 37 & 29 & 27 \end{array} |$

Step 2.4.2

Evaluate the determinant.

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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.

$| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.4.2.1.4

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

$21 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.4.2.1.5

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

$| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.4.2.1.6

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

$- 7 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.4.2.1.7

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

$| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.1.8

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

$22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.1.9

Add the terms together.

$a_{14} = 21 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} | - 7 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.2

Evaluate $| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$ .

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Step 2.4.2.2.1

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

$a_{14} = 21 (8 \cdot 27 - 29 \cdot 26) - 7 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

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 $8$ by $27$ .

$a_{14} = 21 (216 - 29 \cdot 26) - 7 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.2.2.1.2

Multiply $- 29$ by $26$ .

$a_{14} = 21 (216 - 754) - 7 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.2.2.2

Subtract $754$ from $216$ .

$a_{14} = 21 \cdot - 538 - 7 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.3

Evaluate $| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$ .

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Step 2.4.2.3.1

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

$a_{14} = 21 \cdot - 538 - 7 (28 \cdot 27 - 37 \cdot 26) + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

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 $28$ by $27$ .

$a_{14} = 21 \cdot - 538 - 7 (756 - 37 \cdot 26) + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.3.2.1.2

Multiply $- 37$ by $26$ .

$a_{14} = 21 \cdot - 538 - 7 (756 - 962) + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.3.2.2

Subtract $962$ from $756$ .

$a_{14} = 21 \cdot - 538 - 7 \cdot - 206 + 22 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.4.2.4

Evaluate $| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$ .

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Step 2.4.2.4.1

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

$a_{14} = 21 \cdot - 538 - 7 \cdot - 206 + 22 (28 \cdot 29 - 37 \cdot 8)$

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 $28$ by $29$ .

$a_{14} = 21 \cdot - 538 - 7 \cdot - 206 + 22 (812 - 37 \cdot 8)$

Step 2.4.2.4.2.1.2

Multiply $- 37$ by $8$ .

$a_{14} = 21 \cdot - 538 - 7 \cdot - 206 + 22 (812 - 296)$

Step 2.4.2.4.2.2

Subtract $296$ from $812$ .

$a_{14} = 21 \cdot - 538 - 7 \cdot - 206 + 22 \cdot 516$

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 $21$ by $- 538$ .

$a_{14} = - 11298 - 7 \cdot - 206 + 22 \cdot 516$

Step 2.4.2.5.1.2

Multiply $- 7$ by $- 206$ .

$a_{14} = - 11298 + 1442 + 22 \cdot 516$

Step 2.4.2.5.1.3

Multiply $22$ by $516$ .

$a_{14} = - 11298 + 1442 + 11352$

Step 2.4.2.5.2

Add $- 11298$ and $1442$ .

$a_{14} = - 9856 + 11352$

Step 2.4.2.5.3

Add $- 9856$ and $11352$ .

$a_{14} = 1496$

Step 2.5

Calculate the minor for element $a_{21}$ .

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Step 2.5.1

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 6 & 7 & 1 \\ 8 & 26 & 21 \\ 29 & 27 & 24 \end{array} |$

Step 2.5.2

Evaluate the determinant.

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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.

$| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.5.2.1.4

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

$6 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.5.2.1.5

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

$| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.5.2.1.6

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

$- 7 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.5.2.1.7

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

$| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.1.8

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

$1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.1.9

Add the terms together.

$a_{21} = 6 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} | - 7 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.2

Evaluate $| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$ .

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Step 2.5.2.2.1

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

$a_{21} = 6 (26 \cdot 24 - 27 \cdot 21) - 7 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

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 $26$ by $24$ .

$a_{21} = 6 (624 - 27 \cdot 21) - 7 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.2.2.1.2

Multiply $- 27$ by $21$ .

$a_{21} = 6 (624 - 567) - 7 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.2.2.2

Subtract $567$ from $624$ .

$a_{21} = 6 \cdot 57 - 7 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.3

Evaluate $| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$ .

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Step 2.5.2.3.1

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

$a_{21} = 6 \cdot 57 - 7 (8 \cdot 24 - 29 \cdot 21) + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

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 $8$ by $24$ .

$a_{21} = 6 \cdot 57 - 7 (192 - 29 \cdot 21) + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.3.2.1.2

Multiply $- 29$ by $21$ .

$a_{21} = 6 \cdot 57 - 7 (192 - 609) + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.3.2.2

Subtract $609$ from $192$ .

$a_{21} = 6 \cdot 57 - 7 \cdot - 417 + 1 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.5.2.4

Evaluate $| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$ .

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Step 2.5.2.4.1

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

$a_{21} = 6 \cdot 57 - 7 \cdot - 417 + 1 (8 \cdot 27 - 29 \cdot 26)$

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 $8$ by $27$ .

$a_{21} = 6 \cdot 57 - 7 \cdot - 417 + 1 (216 - 29 \cdot 26)$

Step 2.5.2.4.2.1.2

Multiply $- 29$ by $26$ .

$a_{21} = 6 \cdot 57 - 7 \cdot - 417 + 1 (216 - 754)$

Step 2.5.2.4.2.2

Subtract $754$ from $216$ .

$a_{21} = 6 \cdot 57 - 7 \cdot - 417 + 1 \cdot - 538$

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 $6$ by $57$ .

$a_{21} = 342 - 7 \cdot - 417 + 1 \cdot - 538$

Step 2.5.2.5.1.2

Multiply $- 7$ by $- 417$ .

$a_{21} = 342 + 2919 + 1 \cdot - 538$

Step 2.5.2.5.1.3

Multiply $- 538$ by $1$ .

$a_{21} = 342 + 2919 - 538$

Step 2.5.2.5.2

Add $342$ and $2919$ .

$a_{21} = 3261 - 538$

Step 2.5.2.5.3

Subtract $538$ from $3261$ .

$a_{21} = 2723$

Step 2.6

Calculate the minor for element $a_{22}$ .

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Step 2.6.1

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 12 & 7 & 1 \\ 28 & 26 & 21 \\ 37 & 27 & 24 \end{array} |$

Step 2.6.2

Evaluate the determinant.

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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.

$| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.6.2.1.4

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

$12 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$

Step 2.6.2.1.5

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

$| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.6.2.1.6

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

$- 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.6.2.1.7

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

$| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.1.8

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

$1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.1.9

Add the terms together.

$a_{22} = 12 | \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} | - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.2

Evaluate $| \begin{array}{c} 26 & 21 \\ 27 & 24 \end{array} |$ .

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Step 2.6.2.2.1

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

$a_{22} = 12 (26 \cdot 24 - 27 \cdot 21) - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

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 $26$ by $24$ .

$a_{22} = 12 (624 - 27 \cdot 21) - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.2.2.1.2

Multiply $- 27$ by $21$ .

$a_{22} = 12 (624 - 567) - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.2.2.2

Subtract $567$ from $624$ .

$a_{22} = 12 \cdot 57 - 7 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.3

Evaluate $| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$ .

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Step 2.6.2.3.1

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

$a_{22} = 12 \cdot 57 - 7 (28 \cdot 24 - 37 \cdot 21) + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

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 $28$ by $24$ .

$a_{22} = 12 \cdot 57 - 7 (672 - 37 \cdot 21) + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.3.2.1.2

Multiply $- 37$ by $21$ .

$a_{22} = 12 \cdot 57 - 7 (672 - 777) + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.3.2.2

Subtract $777$ from $672$ .

$a_{22} = 12 \cdot 57 - 7 \cdot - 105 + 1 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.6.2.4

Evaluate $| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$ .

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Step 2.6.2.4.1

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

$a_{22} = 12 \cdot 57 - 7 \cdot - 105 + 1 (28 \cdot 27 - 37 \cdot 26)$

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 $28$ by $27$ .

$a_{22} = 12 \cdot 57 - 7 \cdot - 105 + 1 (756 - 37 \cdot 26)$

Step 2.6.2.4.2.1.2

Multiply $- 37$ by $26$ .

$a_{22} = 12 \cdot 57 - 7 \cdot - 105 + 1 (756 - 962)$

Step 2.6.2.4.2.2

Subtract $962$ from $756$ .

$a_{22} = 12 \cdot 57 - 7 \cdot - 105 + 1 \cdot - 206$

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 $12$ by $57$ .

$a_{22} = 684 - 7 \cdot - 105 + 1 \cdot - 206$

Step 2.6.2.5.1.2

Multiply $- 7$ by $- 105$ .

$a_{22} = 684 + 735 + 1 \cdot - 206$

Step 2.6.2.5.1.3

Multiply $- 206$ by $1$ .

$a_{22} = 684 + 735 - 206$

Step 2.6.2.5.2

Add $684$ and $735$ .

$a_{22} = 1419 - 206$

Step 2.6.2.5.3

Subtract $206$ from $1419$ .

$a_{22} = 1213$

Step 2.7

Calculate the minor for element $a_{23}$ .

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Step 2.7.1

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 12 & 6 & 1 \\ 28 & 8 & 21 \\ 37 & 29 & 24 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

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 2.7.2.1.1

Consider the corresponding sign chart.

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

Step 2.7.2.1.2

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

Step 2.7.2.1.3

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

$| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.7.2.1.4

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

$12 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$

Step 2.7.2.1.5

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

$| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.7.2.1.6

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

$- 6 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$

Step 2.7.2.1.7

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

$| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.1.8

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

$1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.1.9

Add the terms together.

$a_{23} = 12 | \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} | - 6 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.2

Evaluate $| \begin{array}{c} 8 & 21 \\ 29 & 24 \end{array} |$ .

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Step 2.7.2.2.1

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

$a_{23} = 12 (8 \cdot 24 - 29 \cdot 21) - 6 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.2.2

Simplify the determinant.

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Step 2.7.2.2.2.1

Simplify each term.

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Step 2.7.2.2.2.1.1

Multiply $8$ by $24$ .

$a_{23} = 12 (192 - 29 \cdot 21) - 6 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.2.2.1.2

Multiply $- 29$ by $21$ .

$a_{23} = 12 (192 - 609) - 6 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.2.2.2

Subtract $609$ from $192$ .

$a_{23} = 12 \cdot - 417 - 6 | \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.3

Evaluate $| \begin{array}{c} 28 & 21 \\ 37 & 24 \end{array} |$ .

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Step 2.7.2.3.1

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

$a_{23} = 12 \cdot - 417 - 6 (28 \cdot 24 - 37 \cdot 21) + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.3.2

Simplify the determinant.

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Step 2.7.2.3.2.1

Simplify each term.

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Step 2.7.2.3.2.1.1

Multiply $28$ by $24$ .

$a_{23} = 12 \cdot - 417 - 6 (672 - 37 \cdot 21) + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.3.2.1.2

Multiply $- 37$ by $21$ .

$a_{23} = 12 \cdot - 417 - 6 (672 - 777) + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.3.2.2

Subtract $777$ from $672$ .

$a_{23} = 12 \cdot - 417 - 6 \cdot - 105 + 1 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.7.2.4

Evaluate $| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$ .

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Step 2.7.2.4.1

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

$a_{23} = 12 \cdot - 417 - 6 \cdot - 105 + 1 (28 \cdot 29 - 37 \cdot 8)$

Step 2.7.2.4.2

Simplify the determinant.

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Step 2.7.2.4.2.1

Simplify each term.

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Step 2.7.2.4.2.1.1

Multiply $28$ by $29$ .

$a_{23} = 12 \cdot - 417 - 6 \cdot - 105 + 1 (812 - 37 \cdot 8)$

Step 2.7.2.4.2.1.2

Multiply $- 37$ by $8$ .

$a_{23} = 12 \cdot - 417 - 6 \cdot - 105 + 1 (812 - 296)$

Step 2.7.2.4.2.2

Subtract $296$ from $812$ .

$a_{23} = 12 \cdot - 417 - 6 \cdot - 105 + 1 \cdot 516$

Step 2.7.2.5

Simplify the determinant.

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Step 2.7.2.5.1

Simplify each term.

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Step 2.7.2.5.1.1

Multiply $12$ by $- 417$ .

$a_{23} = - 5004 - 6 \cdot - 105 + 1 \cdot 516$

Step 2.7.2.5.1.2

Multiply $- 6$ by $- 105$ .

$a_{23} = - 5004 + 630 + 1 \cdot 516$

Step 2.7.2.5.1.3

Multiply $516$ by $1$ .

$a_{23} = - 5004 + 630 + 516$

Step 2.7.2.5.2

Add $- 5004$ and $630$ .

$a_{23} = - 4374 + 516$

Step 2.7.2.5.3

Add $- 4374$ and $516$ .

$a_{23} = - 3858$

Step 2.8

Calculate the minor for element $a_{24}$ .

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Step 2.8.1

The minor for $a_{24}$ is the determinant with row $2$ and column $4$ deleted.

$| \begin{array}{c} 12 & 6 & 7 \\ 28 & 8 & 26 \\ 37 & 29 & 27 \end{array} |$

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

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 2.8.2.1.1

Consider the corresponding sign chart.

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

Step 2.8.2.1.2

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

Step 2.8.2.1.3

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

$| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.8.2.1.4

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

$12 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$

Step 2.8.2.1.5

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

$| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.8.2.1.6

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

$- 6 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$

Step 2.8.2.1.7

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

$| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.1.8

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

$7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.1.9

Add the terms together.

$a_{24} = 12 | \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} | - 6 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.2

Evaluate $| \begin{array}{c} 8 & 26 \\ 29 & 27 \end{array} |$ .

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Step 2.8.2.2.1

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

$a_{24} = 12 (8 \cdot 27 - 29 \cdot 26) - 6 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.2.2

Simplify the determinant.

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Step 2.8.2.2.2.1

Simplify each term.

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Step 2.8.2.2.2.1.1

Multiply $8$ by $27$ .

$a_{24} = 12 (216 - 29 \cdot 26) - 6 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.2.2.1.2

Multiply $- 29$ by $26$ .

$a_{24} = 12 (216 - 754) - 6 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.2.2.2

Subtract $754$ from $216$ .

$a_{24} = 12 \cdot - 538 - 6 | \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.3

Evaluate $| \begin{array}{c} 28 & 26 \\ 37 & 27 \end{array} |$ .

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Step 2.8.2.3.1

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

$a_{24} = 12 \cdot - 538 - 6 (28 \cdot 27 - 37 \cdot 26) + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.3.2

Simplify the determinant.

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Step 2.8.2.3.2.1

Simplify each term.

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Step 2.8.2.3.2.1.1

Multiply $28$ by $27$ .

$a_{24} = 12 \cdot - 538 - 6 (756 - 37 \cdot 26) + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.3.2.1.2

Multiply $- 37$ by $26$ .

$a_{24} = 12 \cdot - 538 - 6 (756 - 962) + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.3.2.2

Subtract $962$ from $756$ .

$a_{24} = 12 \cdot - 538 - 6 \cdot - 206 + 7 | \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$

Step 2.8.2.4

Evaluate $| \begin{array}{c} 28 & 8 \\ 37 & 29 \end{array} |$ .

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Step 2.8.2.4.1

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

$a_{24} = 12 \cdot - 538 - 6 \cdot - 206 + 7 (28 \cdot 29 - 37 \cdot 8)$

Step 2.8.2.4.2

Simplify the determinant.

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Step 2.8.2.4.2.1

Simplify each term.

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Step 2.8.2.4.2.1.1

Multiply $28$ by $29$ .

$a_{24} = 12 \cdot - 538 - 6 \cdot - 206 + 7 (812 - 37 \cdot 8)$

Step 2.8.2.4.2.1.2

Multiply $- 37$ by $8$ .

$a_{24} = 12 \cdot - 538 - 6 \cdot - 206 + 7 (812 - 296)$

Step 2.8.2.4.2.2

Subtract $296$ from $812$ .

$a_{24} = 12 \cdot - 538 - 6 \cdot - 206 + 7 \cdot 516$

Step 2.8.2.5

Simplify the determinant.

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Step 2.8.2.5.1

Simplify each term.

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Step 2.8.2.5.1.1

Multiply $12$ by $- 538$ .

$a_{24} = - 6456 - 6 \cdot - 206 + 7 \cdot 516$

Step 2.8.2.5.1.2

Multiply $- 6$ by $- 206$ .

$a_{24} = - 6456 + 1236 + 7 \cdot 516$

Step 2.8.2.5.1.3

Multiply $7$ by $516$ .

$a_{24} = - 6456 + 1236 + 3612$

Step 2.8.2.5.2

Add $- 6456$ and $1236$ .

$a_{24} = - 5220 + 3612$

Step 2.8.2.5.3

Add $- 5220$ and $3612$ .

$a_{24} = - 1608$

Step 2.9

Calculate the minor for element $a_{31}$ .

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Step 2.9.1

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 6 & 7 & 1 \\ 7 & 22 & 3 \\ 29 & 27 & 24 \end{array} |$

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

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 2.9.2.1.1

Consider the corresponding sign chart.

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

Step 2.9.2.1.2

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

Step 2.9.2.1.3

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

$| \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} |$

Step 2.9.2.1.4

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

$6 | \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} |$

Step 2.9.2.1.5

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

$| \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} |$

Step 2.9.2.1.6

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

$- 7 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} |$

Step 2.9.2.1.7

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

$| \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.1.8

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

$1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.1.9

Add the terms together.

$a_{31} = 6 | \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} | - 7 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.2

Evaluate $| \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} |$ .

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Step 2.9.2.2.1

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

$a_{31} = 6 (22 \cdot 24 - 27 \cdot 3) - 7 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.2.2

Simplify the determinant.

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Step 2.9.2.2.2.1

Simplify each term.

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Step 2.9.2.2.2.1.1

Multiply $22$ by $24$ .

$a_{31} = 6 (528 - 27 \cdot 3) - 7 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.2.2.1.2

Multiply $- 27$ by $3$ .

$a_{31} = 6 (528 - 81) - 7 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.2.2.2

Subtract $81$ from $528$ .

$a_{31} = 6 \cdot 447 - 7 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.3

Evaluate $| \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} |$ .

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Step 2.9.2.3.1

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

$a_{31} = 6 \cdot 447 - 7 (7 \cdot 24 - 29 \cdot 3) + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.3.2

Simplify the determinant.

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Step 2.9.2.3.2.1

Simplify each term.

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Step 2.9.2.3.2.1.1

Multiply $7$ by $24$ .

$a_{31} = 6 \cdot 447 - 7 (168 - 29 \cdot 3) + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.3.2.1.2

Multiply $- 29$ by $3$ .

$a_{31} = 6 \cdot 447 - 7 (168 - 87) + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.3.2.2

Subtract $87$ from $168$ .

$a_{31} = 6 \cdot 447 - 7 \cdot 81 + 1 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.9.2.4

Evaluate $| \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$ .

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Step 2.9.2.4.1

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

$a_{31} = 6 \cdot 447 - 7 \cdot 81 + 1 (7 \cdot 27 - 29 \cdot 22)$

Step 2.9.2.4.2

Simplify the determinant.

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Step 2.9.2.4.2.1

Simplify each term.

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Step 2.9.2.4.2.1.1

Multiply $7$ by $27$ .

$a_{31} = 6 \cdot 447 - 7 \cdot 81 + 1 (189 - 29 \cdot 22)$

Step 2.9.2.4.2.1.2

Multiply $- 29$ by $22$ .

$a_{31} = 6 \cdot 447 - 7 \cdot 81 + 1 (189 - 638)$

Step 2.9.2.4.2.2

Subtract $638$ from $189$ .

$a_{31} = 6 \cdot 447 - 7 \cdot 81 + 1 \cdot - 449$

Step 2.9.2.5

Simplify the determinant.

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Step 2.9.2.5.1

Simplify each term.

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Step 2.9.2.5.1.1

Multiply $6$ by $447$ .

$a_{31} = 2682 - 7 \cdot 81 + 1 \cdot - 449$

Step 2.9.2.5.1.2

Multiply $- 7$ by $81$ .

$a_{31} = 2682 - 567 + 1 \cdot - 449$

Step 2.9.2.5.1.3

Multiply $- 449$ by $1$ .

$a_{31} = 2682 - 567 - 449$

Step 2.9.2.5.2

Subtract $567$ from $2682$ .

$a_{31} = 2115 - 449$

Step 2.9.2.5.3

Subtract $449$ from $2115$ .

$a_{31} = 1666$

Step 2.10

Calculate the minor for element $a_{32}$ .

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Step 2.10.1

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 12 & 7 & 1 \\ 21 & 22 & 3 \\ 37 & 27 & 24 \end{array} |$

Step 2.10.2

Evaluate the determinant.

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Step 2.10.2.1

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 2.10.2.1.1

Consider the corresponding sign chart.

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

Step 2.10.2.1.2

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

Step 2.10.2.1.3

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

$| \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} |$

Step 2.10.2.1.4

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

$12 | \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} |$

Step 2.10.2.1.5

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

$| \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} |$

Step 2.10.2.1.6

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

$- 7 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} |$

Step 2.10.2.1.7

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

$| \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.1.8

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

$1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.1.9

Add the terms together.

$a_{32} = 12 | \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} | - 7 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.2

Evaluate $| \begin{array}{c} 22 & 3 \\ 27 & 24 \end{array} |$ .

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Step 2.10.2.2.1

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

$a_{32} = 12 (22 \cdot 24 - 27 \cdot 3) - 7 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.2.2

Simplify the determinant.

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Step 2.10.2.2.2.1

Simplify each term.

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Step 2.10.2.2.2.1.1

Multiply $22$ by $24$ .

$a_{32} = 12 (528 - 27 \cdot 3) - 7 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.2.2.1.2

Multiply $- 27$ by $3$ .

$a_{32} = 12 (528 - 81) - 7 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.2.2.2

Subtract $81$ from $528$ .

$a_{32} = 12 \cdot 447 - 7 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.3

Evaluate $| \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} |$ .

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Step 2.10.2.3.1

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

$a_{32} = 12 \cdot 447 - 7 (21 \cdot 24 - 37 \cdot 3) + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.3.2

Simplify the determinant.

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Step 2.10.2.3.2.1

Simplify each term.

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Step 2.10.2.3.2.1.1

Multiply $21$ by $24$ .

$a_{32} = 12 \cdot 447 - 7 (504 - 37 \cdot 3) + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.3.2.1.2

Multiply $- 37$ by $3$ .

$a_{32} = 12 \cdot 447 - 7 (504 - 111) + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.3.2.2

Subtract $111$ from $504$ .

$a_{32} = 12 \cdot 447 - 7 \cdot 393 + 1 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.10.2.4

Evaluate $| \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$ .

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Step 2.10.2.4.1

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

$a_{32} = 12 \cdot 447 - 7 \cdot 393 + 1 (21 \cdot 27 - 37 \cdot 22)$

Step 2.10.2.4.2

Simplify the determinant.

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Step 2.10.2.4.2.1

Simplify each term.

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Step 2.10.2.4.2.1.1

Multiply $21$ by $27$ .

$a_{32} = 12 \cdot 447 - 7 \cdot 393 + 1 (567 - 37 \cdot 22)$

Step 2.10.2.4.2.1.2

Multiply $- 37$ by $22$ .

$a_{32} = 12 \cdot 447 - 7 \cdot 393 + 1 (567 - 814)$

Step 2.10.2.4.2.2

Subtract $814$ from $567$ .

$a_{32} = 12 \cdot 447 - 7 \cdot 393 + 1 \cdot - 247$

Step 2.10.2.5

Simplify the determinant.

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Step 2.10.2.5.1

Simplify each term.

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Step 2.10.2.5.1.1

Multiply $12$ by $447$ .

$a_{32} = 5364 - 7 \cdot 393 + 1 \cdot - 247$

Step 2.10.2.5.1.2

Multiply $- 7$ by $393$ .

$a_{32} = 5364 - 2751 + 1 \cdot - 247$

Step 2.10.2.5.1.3

Multiply $- 247$ by $1$ .

$a_{32} = 5364 - 2751 - 247$

Step 2.10.2.5.2

Subtract $2751$ from $5364$ .

$a_{32} = 2613 - 247$

Step 2.10.2.5.3

Subtract $247$ from $2613$ .

$a_{32} = 2366$

Step 2.11

Calculate the minor for element $a_{33}$ .

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Step 2.11.1

The minor for $a_{33}$ is the determinant with row $3$ and column $3$ deleted.

$| \begin{array}{c} 12 & 6 & 1 \\ 21 & 7 & 3 \\ 37 & 29 & 24 \end{array} |$

Step 2.11.2

Evaluate the determinant.

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Step 2.11.2.1

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 2.11.2.1.1

Consider the corresponding sign chart.

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

Step 2.11.2.1.2

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

Step 2.11.2.1.3

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

$| \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} |$

Step 2.11.2.1.4

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

$12 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} |$

Step 2.11.2.1.5

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

$| \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} |$

Step 2.11.2.1.6

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

$- 6 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} |$

Step 2.11.2.1.7

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

$| \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.1.8

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

$1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.1.9

Add the terms together.

$a_{33} = 12 | \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} | - 6 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.2

Evaluate $| \begin{array}{c} 7 & 3 \\ 29 & 24 \end{array} |$ .

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Step 2.11.2.2.1

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

$a_{33} = 12 (7 \cdot 24 - 29 \cdot 3) - 6 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.2.2

Simplify the determinant.

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Step 2.11.2.2.2.1

Simplify each term.

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Step 2.11.2.2.2.1.1

Multiply $7$ by $24$ .

$a_{33} = 12 (168 - 29 \cdot 3) - 6 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.2.2.1.2

Multiply $- 29$ by $3$ .

$a_{33} = 12 (168 - 87) - 6 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.2.2.2

Subtract $87$ from $168$ .

$a_{33} = 12 \cdot 81 - 6 | \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.3

Evaluate $| \begin{array}{c} 21 & 3 \\ 37 & 24 \end{array} |$ .

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Step 2.11.2.3.1

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

$a_{33} = 12 \cdot 81 - 6 (21 \cdot 24 - 37 \cdot 3) + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.3.2

Simplify the determinant.

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Step 2.11.2.3.2.1

Simplify each term.

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Step 2.11.2.3.2.1.1

Multiply $21$ by $24$ .

$a_{33} = 12 \cdot 81 - 6 (504 - 37 \cdot 3) + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.3.2.1.2

Multiply $- 37$ by $3$ .

$a_{33} = 12 \cdot 81 - 6 (504 - 111) + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.3.2.2

Subtract $111$ from $504$ .

$a_{33} = 12 \cdot 81 - 6 \cdot 393 + 1 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.11.2.4

Evaluate $| \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$ .

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Step 2.11.2.4.1

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

$a_{33} = 12 \cdot 81 - 6 \cdot 393 + 1 (21 \cdot 29 - 37 \cdot 7)$

Step 2.11.2.4.2

Simplify the determinant.

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Step 2.11.2.4.2.1

Simplify each term.

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Step 2.11.2.4.2.1.1

Multiply $21$ by $29$ .

$a_{33} = 12 \cdot 81 - 6 \cdot 393 + 1 (609 - 37 \cdot 7)$

Step 2.11.2.4.2.1.2

Multiply $- 37$ by $7$ .

$a_{33} = 12 \cdot 81 - 6 \cdot 393 + 1 (609 - 259)$

Step 2.11.2.4.2.2

Subtract $259$ from $609$ .

$a_{33} = 12 \cdot 81 - 6 \cdot 393 + 1 \cdot 350$

Step 2.11.2.5

Simplify the determinant.

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Step 2.11.2.5.1

Simplify each term.

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Step 2.11.2.5.1.1

Multiply $12$ by $81$ .

$a_{33} = 972 - 6 \cdot 393 + 1 \cdot 350$

Step 2.11.2.5.1.2

Multiply $- 6$ by $393$ .

$a_{33} = 972 - 2358 + 1 \cdot 350$

Step 2.11.2.5.1.3

Multiply $350$ by $1$ .

$a_{33} = 972 - 2358 + 350$

Step 2.11.2.5.2

Subtract $2358$ from $972$ .

$a_{33} = - 1386 + 350$

Step 2.11.2.5.3

Add $- 1386$ and $350$ .

$a_{33} = - 1036$

Step 2.12

Calculate the minor for element $a_{34}$ .

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Step 2.12.1

The minor for $a_{34}$ is the determinant with row $3$ and column $4$ deleted.

$| \begin{array}{c} 12 & 6 & 7 \\ 21 & 7 & 22 \\ 37 & 29 & 27 \end{array} |$

Step 2.12.2

Evaluate the determinant.

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Step 2.12.2.1

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 2.12.2.1.1

Consider the corresponding sign chart.

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

Step 2.12.2.1.2

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

Step 2.12.2.1.3

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

$| \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.12.2.1.4

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

$12 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$

Step 2.12.2.1.5

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

$| \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.12.2.1.6

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

$- 6 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$

Step 2.12.2.1.7

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

$| \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.1.8

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

$7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.1.9

Add the terms together.

$a_{34} = 12 | \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} | - 6 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.2

Evaluate $| \begin{array}{c} 7 & 22 \\ 29 & 27 \end{array} |$ .

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Step 2.12.2.2.1

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

$a_{34} = 12 (7 \cdot 27 - 29 \cdot 22) - 6 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.2.2

Simplify the determinant.

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Step 2.12.2.2.2.1

Simplify each term.

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Step 2.12.2.2.2.1.1

Multiply $7$ by $27$ .

$a_{34} = 12 (189 - 29 \cdot 22) - 6 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.2.2.1.2

Multiply $- 29$ by $22$ .

$a_{34} = 12 (189 - 638) - 6 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.2.2.2

Subtract $638$ from $189$ .

$a_{34} = 12 \cdot - 449 - 6 | \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.3

Evaluate $| \begin{array}{c} 21 & 22 \\ 37 & 27 \end{array} |$ .

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Step 2.12.2.3.1

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

$a_{34} = 12 \cdot - 449 - 6 (21 \cdot 27 - 37 \cdot 22) + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.3.2

Simplify the determinant.

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Step 2.12.2.3.2.1

Simplify each term.

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Step 2.12.2.3.2.1.1

Multiply $21$ by $27$ .

$a_{34} = 12 \cdot - 449 - 6 (567 - 37 \cdot 22) + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.3.2.1.2

Multiply $- 37$ by $22$ .

$a_{34} = 12 \cdot - 449 - 6 (567 - 814) + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.3.2.2

Subtract $814$ from $567$ .

$a_{34} = 12 \cdot - 449 - 6 \cdot - 247 + 7 | \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$

Step 2.12.2.4

Evaluate $| \begin{array}{c} 21 & 7 \\ 37 & 29 \end{array} |$ .

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Step 2.12.2.4.1

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

$a_{34} = 12 \cdot - 449 - 6 \cdot - 247 + 7 (21 \cdot 29 - 37 \cdot 7)$

Step 2.12.2.4.2

Simplify the determinant.

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Step 2.12.2.4.2.1

Simplify each term.

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Step 2.12.2.4.2.1.1

Multiply $21$ by $29$ .

$a_{34} = 12 \cdot - 449 - 6 \cdot - 247 + 7 (609 - 37 \cdot 7)$

Step 2.12.2.4.2.1.2

Multiply $- 37$ by $7$ .

$a_{34} = 12 \cdot - 449 - 6 \cdot - 247 + 7 (609 - 259)$

Step 2.12.2.4.2.2

Subtract $259$ from $609$ .

$a_{34} = 12 \cdot - 449 - 6 \cdot - 247 + 7 \cdot 350$

Step 2.12.2.5

Simplify the determinant.

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Step 2.12.2.5.1

Simplify each term.

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Step 2.12.2.5.1.1

Multiply $12$ by $- 449$ .

$a_{34} = - 5388 - 6 \cdot - 247 + 7 \cdot 350$

Step 2.12.2.5.1.2

Multiply $- 6$ by $- 247$ .

$a_{34} = - 5388 + 1482 + 7 \cdot 350$

Step 2.12.2.5.1.3

Multiply $7$ by $350$ .

$a_{34} = - 5388 + 1482 + 2450$

Step 2.12.2.5.2

Add $- 5388$ and $1482$ .

$a_{34} = - 3906 + 2450$

Step 2.12.2.5.3

Add $- 3906$ and $2450$ .

$a_{34} = - 1456$

Step 2.13

Calculate the minor for element $a_{41}$ .

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Step 2.13.1

The minor for $a_{41}$ is the determinant with row $4$ and column $1$ deleted.

$| \begin{array}{c} 6 & 7 & 1 \\ 7 & 22 & 3 \\ 8 & 26 & 21 \end{array} |$

Step 2.13.2

Evaluate the determinant.

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Step 2.13.2.1

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 2.13.2.1.1

Consider the corresponding sign chart.

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

Step 2.13.2.1.2

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

Step 2.13.2.1.3

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

$| \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} |$

Step 2.13.2.1.4

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

$6 | \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} |$

Step 2.13.2.1.5

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

$| \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} |$

Step 2.13.2.1.6

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

$- 7 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} |$

Step 2.13.2.1.7

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

$| \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.1.8

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

$1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.1.9

Add the terms together.

$a_{41} = 6 | \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} | - 7 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.2

Evaluate $| \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} |$ .

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Step 2.13.2.2.1

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

$a_{41} = 6 (22 \cdot 21 - 26 \cdot 3) - 7 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.2.2

Simplify the determinant.

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Step 2.13.2.2.2.1

Simplify each term.

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Step 2.13.2.2.2.1.1

Multiply $22$ by $21$ .

$a_{41} = 6 (462 - 26 \cdot 3) - 7 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.2.2.1.2

Multiply $- 26$ by $3$ .

$a_{41} = 6 (462 - 78) - 7 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.2.2.2

Subtract $78$ from $462$ .

$a_{41} = 6 \cdot 384 - 7 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} | + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.3

Evaluate $| \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} |$ .

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Step 2.13.2.3.1

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

$a_{41} = 6 \cdot 384 - 7 (7 \cdot 21 - 8 \cdot 3) + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.3.2

Simplify the determinant.

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Step 2.13.2.3.2.1

Simplify each term.

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Step 2.13.2.3.2.1.1

Multiply $7$ by $21$ .

$a_{41} = 6 \cdot 384 - 7 (147 - 8 \cdot 3) + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.3.2.1.2

Multiply $- 8$ by $3$ .

$a_{41} = 6 \cdot 384 - 7 (147 - 24) + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.3.2.2

Subtract $24$ from $147$ .

$a_{41} = 6 \cdot 384 - 7 \cdot 123 + 1 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.13.2.4

Evaluate $| \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$ .

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Step 2.13.2.4.1

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

$a_{41} = 6 \cdot 384 - 7 \cdot 123 + 1 (7 \cdot 26 - 8 \cdot 22)$

Step 2.13.2.4.2

Simplify the determinant.

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Step 2.13.2.4.2.1

Simplify each term.

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Step 2.13.2.4.2.1.1

Multiply $7$ by $26$ .

$a_{41} = 6 \cdot 384 - 7 \cdot 123 + 1 (182 - 8 \cdot 22)$

Step 2.13.2.4.2.1.2

Multiply $- 8$ by $22$ .

$a_{41} = 6 \cdot 384 - 7 \cdot 123 + 1 (182 - 176)$

Step 2.13.2.4.2.2

Subtract $176$ from $182$ .

$a_{41} = 6 \cdot 384 - 7 \cdot 123 + 1 \cdot 6$

Step 2.13.2.5

Simplify the determinant.

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Step 2.13.2.5.1

Simplify each term.

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Step 2.13.2.5.1.1

Multiply $6$ by $384$ .

$a_{41} = 2304 - 7 \cdot 123 + 1 \cdot 6$

Step 2.13.2.5.1.2

Multiply $- 7$ by $123$ .

$a_{41} = 2304 - 861 + 1 \cdot 6$

Step 2.13.2.5.1.3

Multiply $6$ by $1$ .

$a_{41} = 2304 - 861 + 6$

Step 2.13.2.5.2

Subtract $861$ from $2304$ .

$a_{41} = 1443 + 6$

Step 2.13.2.5.3

Add $1443$ and $6$ .

$a_{41} = 1449$

Step 2.14

Calculate the minor for element $a_{42}$ .

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Step 2.14.1

The minor for $a_{42}$ is the determinant with row $4$ and column $2$ deleted.

$| \begin{array}{c} 12 & 7 & 1 \\ 21 & 22 & 3 \\ 28 & 26 & 21 \end{array} |$

Step 2.14.2

Evaluate the determinant.

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Step 2.14.2.1

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 2.14.2.1.1

Consider the corresponding sign chart.

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

Step 2.14.2.1.2

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

Step 2.14.2.1.3

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

$| \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} |$

Step 2.14.2.1.4

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

$12 | \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} |$

Step 2.14.2.1.5

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

$| \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} |$

Step 2.14.2.1.6

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

$- 7 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} |$

Step 2.14.2.1.7

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

$| \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.1.8

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

$1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.1.9

Add the terms together.

$a_{42} = 12 | \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} | - 7 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.2

Evaluate $| \begin{array}{c} 22 & 3 \\ 26 & 21 \end{array} |$ .

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Step 2.14.2.2.1

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

$a_{42} = 12 (22 \cdot 21 - 26 \cdot 3) - 7 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.2.2

Simplify the determinant.

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Step 2.14.2.2.2.1

Simplify each term.

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Step 2.14.2.2.2.1.1

Multiply $22$ by $21$ .

$a_{42} = 12 (462 - 26 \cdot 3) - 7 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.2.2.1.2

Multiply $- 26$ by $3$ .

$a_{42} = 12 (462 - 78) - 7 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.2.2.2

Subtract $78$ from $462$ .

$a_{42} = 12 \cdot 384 - 7 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.3

Evaluate $| \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} |$ .

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Step 2.14.2.3.1

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

$a_{42} = 12 \cdot 384 - 7 (21 \cdot 21 - 28 \cdot 3) + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.3.2

Simplify the determinant.

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Step 2.14.2.3.2.1

Simplify each term.

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Step 2.14.2.3.2.1.1

Multiply $21$ by $21$ .

$a_{42} = 12 \cdot 384 - 7 (441 - 28 \cdot 3) + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.3.2.1.2

Multiply $- 28$ by $3$ .

$a_{42} = 12 \cdot 384 - 7 (441 - 84) + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.3.2.2

Subtract $84$ from $441$ .

$a_{42} = 12 \cdot 384 - 7 \cdot 357 + 1 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.14.2.4

Evaluate $| \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$ .

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Step 2.14.2.4.1

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

$a_{42} = 12 \cdot 384 - 7 \cdot 357 + 1 (21 \cdot 26 - 28 \cdot 22)$

Step 2.14.2.4.2

Simplify the determinant.

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Step 2.14.2.4.2.1

Simplify each term.

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Step 2.14.2.4.2.1.1

Multiply $21$ by $26$ .

$a_{42} = 12 \cdot 384 - 7 \cdot 357 + 1 (546 - 28 \cdot 22)$

Step 2.14.2.4.2.1.2

Multiply $- 28$ by $22$ .

$a_{42} = 12 \cdot 384 - 7 \cdot 357 + 1 (546 - 616)$

Step 2.14.2.4.2.2

Subtract $616$ from $546$ .

$a_{42} = 12 \cdot 384 - 7 \cdot 357 + 1 \cdot - 70$

Step 2.14.2.5

Simplify the determinant.

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Step 2.14.2.5.1

Simplify each term.

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Step 2.14.2.5.1.1

Multiply $12$ by $384$ .

$a_{42} = 4608 - 7 \cdot 357 + 1 \cdot - 70$

Step 2.14.2.5.1.2

Multiply $- 7$ by $357$ .

$a_{42} = 4608 - 2499 + 1 \cdot - 70$

Step 2.14.2.5.1.3

Multiply $- 70$ by $1$ .

$a_{42} = 4608 - 2499 - 70$

Step 2.14.2.5.2

Subtract $2499$ from $4608$ .

$a_{42} = 2109 - 70$

Step 2.14.2.5.3

Subtract $70$ from $2109$ .

$a_{42} = 2039$

Step 2.15

Calculate the minor for element $a_{43}$ .

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Step 2.15.1

The minor for $a_{43}$ is the determinant with row $4$ and column $3$ deleted.

$| \begin{array}{c} 12 & 6 & 1 \\ 21 & 7 & 3 \\ 28 & 8 & 21 \end{array} |$

Step 2.15.2

Evaluate the determinant.

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Step 2.15.2.1

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 2.15.2.1.1

Consider the corresponding sign chart.

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

Step 2.15.2.1.2

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

Step 2.15.2.1.3

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

$| \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} |$

Step 2.15.2.1.4

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

$12 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} |$

Step 2.15.2.1.5

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

$| \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} |$

Step 2.15.2.1.6

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

$- 6 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} |$

Step 2.15.2.1.7

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

$| \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.1.8

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

$1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.1.9

Add the terms together.

$a_{43} = 12 | \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} | - 6 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.2

Evaluate $| \begin{array}{c} 7 & 3 \\ 8 & 21 \end{array} |$ .

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Step 2.15.2.2.1

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

$a_{43} = 12 (7 \cdot 21 - 8 \cdot 3) - 6 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.2.2

Simplify the determinant.

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Step 2.15.2.2.2.1

Simplify each term.

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Step 2.15.2.2.2.1.1

Multiply $7$ by $21$ .

$a_{43} = 12 (147 - 8 \cdot 3) - 6 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.2.2.1.2

Multiply $- 8$ by $3$ .

$a_{43} = 12 (147 - 24) - 6 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.2.2.2

Subtract $24$ from $147$ .

$a_{43} = 12 \cdot 123 - 6 | \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} | + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.3

Evaluate $| \begin{array}{c} 21 & 3 \\ 28 & 21 \end{array} |$ .

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Step 2.15.2.3.1

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

$a_{43} = 12 \cdot 123 - 6 (21 \cdot 21 - 28 \cdot 3) + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.3.2

Simplify the determinant.

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Step 2.15.2.3.2.1

Simplify each term.

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Step 2.15.2.3.2.1.1

Multiply $21$ by $21$ .

$a_{43} = 12 \cdot 123 - 6 (441 - 28 \cdot 3) + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.3.2.1.2

Multiply $- 28$ by $3$ .

$a_{43} = 12 \cdot 123 - 6 (441 - 84) + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.3.2.2

Subtract $84$ from $441$ .

$a_{43} = 12 \cdot 123 - 6 \cdot 357 + 1 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.15.2.4

Evaluate $| \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$ .

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Step 2.15.2.4.1

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

$a_{43} = 12 \cdot 123 - 6 \cdot 357 + 1 (21 \cdot 8 - 28 \cdot 7)$

Step 2.15.2.4.2

Simplify the determinant.

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Step 2.15.2.4.2.1

Simplify each term.

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Step 2.15.2.4.2.1.1

Multiply $21$ by $8$ .

$a_{43} = 12 \cdot 123 - 6 \cdot 357 + 1 (168 - 28 \cdot 7)$

Step 2.15.2.4.2.1.2

Multiply $- 28$ by $7$ .

$a_{43} = 12 \cdot 123 - 6 \cdot 357 + 1 (168 - 196)$

Step 2.15.2.4.2.2

Subtract $196$ from $168$ .

$a_{43} = 12 \cdot 123 - 6 \cdot 357 + 1 \cdot - 28$

Step 2.15.2.5

Simplify the determinant.

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Step 2.15.2.5.1

Simplify each term.

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Step 2.15.2.5.1.1

Multiply $12$ by $123$ .

$a_{43} = 1476 - 6 \cdot 357 + 1 \cdot - 28$

Step 2.15.2.5.1.2

Multiply $- 6$ by $357$ .

$a_{43} = 1476 - 2142 + 1 \cdot - 28$

Step 2.15.2.5.1.3

Multiply $- 28$ by $1$ .

$a_{43} = 1476 - 2142 - 28$

Step 2.15.2.5.2

Subtract $2142$ from $1476$ .

$a_{43} = - 666 - 28$

Step 2.15.2.5.3

Subtract $28$ from $- 666$ .

$a_{43} = - 694$

Step 2.16

Calculate the minor for element $a_{44}$ .

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Step 2.16.1

The minor for $a_{44}$ is the determinant with row $4$ and column $4$ deleted.

$| \begin{array}{c} 12 & 6 & 7 \\ 21 & 7 & 22 \\ 28 & 8 & 26 \end{array} |$

Step 2.16.2

Evaluate the determinant.

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Step 2.16.2.1

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 2.16.2.1.1

Consider the corresponding sign chart.

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

Step 2.16.2.1.2

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

Step 2.16.2.1.3

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

$| \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.16.2.1.4

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

$12 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$

Step 2.16.2.1.5

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

$| \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.16.2.1.6

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

$- 6 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$

Step 2.16.2.1.7

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

$| \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.1.8

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

$7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.1.9

Add the terms together.

$a_{44} = 12 | \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} | - 6 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.2

Evaluate $| \begin{array}{c} 7 & 22 \\ 8 & 26 \end{array} |$ .

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Step 2.16.2.2.1

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

$a_{44} = 12 (7 \cdot 26 - 8 \cdot 22) - 6 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.2.2

Simplify the determinant.

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Step 2.16.2.2.2.1

Simplify each term.

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Step 2.16.2.2.2.1.1

Multiply $7$ by $26$ .

$a_{44} = 12 (182 - 8 \cdot 22) - 6 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.2.2.1.2

Multiply $- 8$ by $22$ .

$a_{44} = 12 (182 - 176) - 6 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.2.2.2

Subtract $176$ from $182$ .

$a_{44} = 12 \cdot 6 - 6 | \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} | + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.3

Evaluate $| \begin{array}{c} 21 & 22 \\ 28 & 26 \end{array} |$ .

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Step 2.16.2.3.1

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

$a_{44} = 12 \cdot 6 - 6 (21 \cdot 26 - 28 \cdot 22) + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.3.2

Simplify the determinant.

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Step 2.16.2.3.2.1

Simplify each term.

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Step 2.16.2.3.2.1.1

Multiply $21$ by $26$ .

$a_{44} = 12 \cdot 6 - 6 (546 - 28 \cdot 22) + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.3.2.1.2

Multiply $- 28$ by $22$ .

$a_{44} = 12 \cdot 6 - 6 (546 - 616) + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.3.2.2

Subtract $616$ from $546$ .

$a_{44} = 12 \cdot 6 - 6 \cdot - 70 + 7 | \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$

Step 2.16.2.4

Evaluate $| \begin{array}{c} 21 & 7 \\ 28 & 8 \end{array} |$ .

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Step 2.16.2.4.1

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

$a_{44} = 12 \cdot 6 - 6 \cdot - 70 + 7 (21 \cdot 8 - 28 \cdot 7)$

Step 2.16.2.4.2

Simplify the determinant.

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Step 2.16.2.4.2.1

Simplify each term.

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Step 2.16.2.4.2.1.1

Multiply $21$ by $8$ .

$a_{44} = 12 \cdot 6 - 6 \cdot - 70 + 7 (168 - 28 \cdot 7)$

Step 2.16.2.4.2.1.2

Multiply $- 28$ by $7$ .

$a_{44} = 12 \cdot 6 - 6 \cdot - 70 + 7 (168 - 196)$

Step 2.16.2.4.2.2

Subtract $196$ from $168$ .

$a_{44} = 12 \cdot 6 - 6 \cdot - 70 + 7 \cdot - 28$

Step 2.16.2.5

Simplify the determinant.

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Step 2.16.2.5.1

Simplify each term.

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Step 2.16.2.5.1.1

Multiply $12$ by $6$ .

$a_{44} = 72 - 6 \cdot - 70 + 7 \cdot - 28$

Step 2.16.2.5.1.2

Multiply $- 6$ by $- 70$ .

$a_{44} = 72 + 420 + 7 \cdot - 28$

Step 2.16.2.5.1.3

Multiply $7$ by $- 28$ .

$a_{44} = 72 + 420 - 196$

Step 2.16.2.5.2

Add $72$ and $420$ .

$a_{44} = 492 - 196$

Step 2.16.2.5.3

Subtract $196$ from $492$ .

$a_{44} = 296$

Step 2.17

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the $-$ positions on the sign chart.

$[\begin{array}{c} 7959 & - 2889 & - 6474 & - 1496 \\ - 2723 & 1213 & 3858 & - 1608 \\ 1666 & - 2366 & - 1036 & 1456 \\ - 1449 & 2039 & 694 & 296 \end{array}]$

Step 3

Transpose the matrix by switching its rows to columns.

$[\begin{array}{c} 7959 & - 2723 & 1666 & - 1449 \\ - 2889 & 1213 & - 2366 & 2039 \\ - 6474 & 3858 & - 1036 & 694 \\ - 1496 & - 1608 & 1456 & 296 \end{array}]$