Linear Algebra Examples

$[\begin{array}{c} 2 & 5 & 2 & 2 \\ 2 & 5 & 2 & 2 \\ 1 & 21 & 2 & 2 \\ 1 & 1 & 1 & 1 \end{array}] \cdot [\begin{array}{c} 5 & 7 & 7 & 10 \\ 3 & 3 & 2 & 1 \\ 3 & 47 & 58 & 5 \\ 4 & 4 & 10 & 10 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 & 5 & 2 & 2 \\ 2 & 5 & 2 & 2 \\ 1 & 21 & 2 & 2 \\ 1 & 1 & 1 & 1 \end{array}] [\begin{array}{c} 5 & 7 & 7 & 10 \\ 3 & 3 & 2 & 1 \\ 3 & 47 & 58 & 5 \\ 4 & 4 & 10 & 10 \end{array}]$ .

Tap for more steps...

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 $4 \times 4$ and the second matrix is $4 \times 4$ .

Step 1.2

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

$[\begin{array}{c} 2 \cdot 5 + 5 \cdot 3 + 2 \cdot 3 + 2 \cdot 4 & 2 \cdot 7 + 5 \cdot 3 + 2 \cdot 47 + 2 \cdot 4 & 2 \cdot 7 + 5 \cdot 2 + 2 \cdot 58 + 2 \cdot 10 & 2 \cdot 10 + 5 \cdot 1 + 2 \cdot 5 + 2 \cdot 10 \\ 2 \cdot 5 + 5 \cdot 3 + 2 \cdot 3 + 2 \cdot 4 & 2 \cdot 7 + 5 \cdot 3 + 2 \cdot 47 + 2 \cdot 4 & 2 \cdot 7 + 5 \cdot 2 + 2 \cdot 58 + 2 \cdot 10 & 2 \cdot 10 + 5 \cdot 1 + 2 \cdot 5 + 2 \cdot 10 \\ 1 \cdot 5 + 21 \cdot 3 + 2 \cdot 3 + 2 \cdot 4 & 1 \cdot 7 + 21 \cdot 3 + 2 \cdot 47 + 2 \cdot 4 & 1 \cdot 7 + 21 \cdot 2 + 2 \cdot 58 + 2 \cdot 10 & 1 \cdot 10 + 21 \cdot 1 + 2 \cdot 5 + 2 \cdot 10 \\ 1 \cdot 5 + 1 \cdot 3 + 1 \cdot 3 + 1 \cdot 4 & 1 \cdot 7 + 1 \cdot 3 + 1 \cdot 47 + 1 \cdot 4 & 1 \cdot 7 + 1 \cdot 2 + 1 \cdot 58 + 1 \cdot 10 & 1 \cdot 10 + 1 \cdot 1 + 1 \cdot 5 + 1 \cdot 10 \end{array}]$

Step 1.3

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

$[\begin{array}{c} 39 & 131 & 160 & 55 \\ 39 & 131 & 160 & 55 \\ 82 & 172 & 185 & 61 \\ 15 & 61 & 77 & 26 \end{array}]$

Step 2

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.

Tap for more steps...

Step 2.1

Consider the corresponding sign chart.

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

Step 2.2

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

Step 2.3

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

$| \begin{array}{c} 131 & 160 & 55 \\ 172 & 185 & 61 \\ 61 & 77 & 26 \end{array} |$

Step 2.4

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

$39 | \begin{array}{c} 131 & 160 & 55 \\ 172 & 185 & 61 \\ 61 & 77 & 26 \end{array} |$

Step 2.5

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

$| \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} |$

Step 2.6

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

$- 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} |$

Step 2.7

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

$| \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} |$

Step 2.8

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

$160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} |$

Step 2.9

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

$| \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 2.10

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

$- 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 2.11

Add the terms together.

$39 | \begin{array}{c} 131 & 160 & 55 \\ 172 & 185 & 61 \\ 61 & 77 & 26 \end{array} | - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 131 & 160 & 55 \\ 172 & 185 & 61 \\ 61 & 77 & 26 \end{array} |$ .

Tap for more steps...

Step 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.

Tap for more steps...

Step 3.1.1

Consider the corresponding sign chart.

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

Step 3.1.2

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

Step 3.1.3

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

$| \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} |$

Step 3.1.4

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

$131 | \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} |$

Step 3.1.5

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

$| \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} |$

Step 3.1.6

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

$- 160 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} |$

Step 3.1.7

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

$| \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |$

Step 3.1.8

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

$55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |$

Step 3.1.9

Add the terms together.

$39 (131 | \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} | - 160 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} | + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.2

Evaluate $| \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} |$ .

Tap for more steps...

Step 3.2.1

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

$39 (131 (185 \cdot 26 - 77 \cdot 61) - 160 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} | + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.2.2

Simplify the determinant.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Multiply $185$ by $26$ .

$39 (131 (4810 - 77 \cdot 61) - 160 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} | + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.2.2.1.2

Multiply $- 77$ by $61$ .

$39 (131 (4810 - 4697) - 160 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} | + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.2.2.2

Subtract $4697$ from $4810$ .

$39 (131 \cdot 113 - 160 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} | + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.3

Evaluate $| \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} |$ .

Tap for more steps...

Step 3.3.1

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

$39 (131 \cdot 113 - 160 (172 \cdot 26 - 61 \cdot 61) + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.3.2

Simplify the determinant.

Tap for more steps...

Step 3.3.2.1

Simplify each term.

Tap for more steps...

Step 3.3.2.1.1

Multiply $172$ by $26$ .

$39 (131 \cdot 113 - 160 (4472 - 61 \cdot 61) + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.3.2.1.2

Multiply $- 61$ by $61$ .

$39 (131 \cdot 113 - 160 (4472 - 3721) + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.3.2.2

Subtract $3721$ from $4472$ .

$39 (131 \cdot 113 - 160 \cdot 751 + 55 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.4

Evaluate $| \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |$ .

Tap for more steps...

Step 3.4.1

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

$39 (131 \cdot 113 - 160 \cdot 751 + 55 (172 \cdot 77 - 61 \cdot 185)) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.4.2

Simplify the determinant.

Tap for more steps...

Step 3.4.2.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1

Multiply $172$ by $77$ .

$39 (131 \cdot 113 - 160 \cdot 751 + 55 (13244 - 61 \cdot 185)) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.4.2.1.2

Multiply $- 61$ by $185$ .

$39 (131 \cdot 113 - 160 \cdot 751 + 55 (13244 - 11285)) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.4.2.2

Subtract $11285$ from $13244$ .

$39 (131 \cdot 113 - 160 \cdot 751 + 55 \cdot 1959) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.5

Simplify the determinant.

Tap for more steps...

Step 3.5.1

Simplify each term.

Tap for more steps...

Step 3.5.1.1

Multiply $131$ by $113$ .

$39 (14803 - 160 \cdot 751 + 55 \cdot 1959) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.5.1.2

Multiply $- 160$ by $751$ .

$39 (14803 - 120160 + 55 \cdot 1959) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.5.1.3

Multiply $55$ by $1959$ .

$39 (14803 - 120160 + 107745) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.5.2

Subtract $120160$ from $14803$ .

$39 (- 105357 + 107745) - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 3.5.3

Add $- 105357$ and $107745$ .

$39 \cdot 2388 - 131 | \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} | + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 39 & 160 & 55 \\ 82 & 185 & 61 \\ 15 & 77 & 26 \end{array} |$ .

Tap for more steps...

Step 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.

Tap for more steps...

Step 4.1.1

Consider the corresponding sign chart.

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

Step 4.1.2

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

Step 4.1.3

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

$| \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} |$

Step 4.1.4

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

$39 | \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} |$

Step 4.1.5

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

$| \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} |$

Step 4.1.6

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

$- 160 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} |$

Step 4.1.7

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

$| \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |$

Step 4.1.8

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

$55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |$

Step 4.1.9

Add the terms together.

$39 \cdot 2388 - 131 (39 | \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} | - 160 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.2

Evaluate $| \begin{array}{c} 185 & 61 \\ 77 & 26 \end{array} |$ .

Tap for more steps...

Step 4.2.1

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

$39 \cdot 2388 - 131 (39 (185 \cdot 26 - 77 \cdot 61) - 160 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.2.2

Simplify the determinant.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Multiply $185$ by $26$ .

$39 \cdot 2388 - 131 (39 (4810 - 77 \cdot 61) - 160 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.2.2.1.2

Multiply $- 77$ by $61$ .

$39 \cdot 2388 - 131 (39 (4810 - 4697) - 160 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.2.2.2

Subtract $4697$ from $4810$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.3

Evaluate $| \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} |$ .

Tap for more steps...

Step 4.3.1

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

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 (82 \cdot 26 - 15 \cdot 61) + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.3.2

Simplify the determinant.

Tap for more steps...

Step 4.3.2.1

Simplify each term.

Tap for more steps...

Step 4.3.2.1.1

Multiply $82$ by $26$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 (2132 - 15 \cdot 61) + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.3.2.1.2

Multiply $- 15$ by $61$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 (2132 - 915) + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.3.2.2

Subtract $915$ from $2132$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 \cdot 1217 + 55 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.4

Evaluate $| \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |$ .

Tap for more steps...

Step 4.4.1

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

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 \cdot 1217 + 55 (82 \cdot 77 - 15 \cdot 185)) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.4.2

Simplify the determinant.

Tap for more steps...

Step 4.4.2.1

Simplify each term.

Tap for more steps...

Step 4.4.2.1.1

Multiply $82$ by $77$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 \cdot 1217 + 55 (6314 - 15 \cdot 185)) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.4.2.1.2

Multiply $- 15$ by $185$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 \cdot 1217 + 55 (6314 - 2775)) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.4.2.2

Subtract $2775$ from $6314$ .

$39 \cdot 2388 - 131 (39 \cdot 113 - 160 \cdot 1217 + 55 \cdot 3539) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.5

Simplify the determinant.

Tap for more steps...

Step 4.5.1

Simplify each term.

Tap for more steps...

Step 4.5.1.1

Multiply $39$ by $113$ .

$39 \cdot 2388 - 131 (4407 - 160 \cdot 1217 + 55 \cdot 3539) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.5.1.2

Multiply $- 160$ by $1217$ .

$39 \cdot 2388 - 131 (4407 - 194720 + 55 \cdot 3539) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.5.1.3

Multiply $55$ by $3539$ .

$39 \cdot 2388 - 131 (4407 - 194720 + 194645) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.5.2

Subtract $194720$ from $4407$ .

$39 \cdot 2388 - 131 (- 190313 + 194645) + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 4.5.3

Add $- 190313$ and $194645$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 | \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} | - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 39 & 131 & 55 \\ 82 & 172 & 61 \\ 15 & 61 & 26 \end{array} |$ .

Tap for more steps...

Step 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.

Tap for more steps...

Step 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

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

Step 5.1.3

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

$| \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} |$

Step 5.1.4

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

$39 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} |$

Step 5.1.6

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

$- 131 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |$

Step 5.1.8

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

$55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |$

Step 5.1.9

Add the terms together.

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 | \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} | - 131 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.2

Evaluate $| \begin{array}{c} 172 & 61 \\ 61 & 26 \end{array} |$ .

Tap for more steps...

Step 5.2.1

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

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 (172 \cdot 26 - 61 \cdot 61) - 131 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.2.2

Simplify the determinant.

Tap for more steps...

Step 5.2.2.1

Simplify each term.

Tap for more steps...

Step 5.2.2.1.1

Multiply $172$ by $26$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 (4472 - 61 \cdot 61) - 131 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.2.2.1.2

Multiply $- 61$ by $61$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 (4472 - 3721) - 131 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.2.2.2

Subtract $3721$ from $4472$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 | \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} | + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.3

Evaluate $| \begin{array}{c} 82 & 61 \\ 15 & 26 \end{array} |$ .

Tap for more steps...

Step 5.3.1

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

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 (82 \cdot 26 - 15 \cdot 61) + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.3.2

Simplify the determinant.

Tap for more steps...

Step 5.3.2.1

Simplify each term.

Tap for more steps...

Step 5.3.2.1.1

Multiply $82$ by $26$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 (2132 - 15 \cdot 61) + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.3.2.1.2

Multiply $- 15$ by $61$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 (2132 - 915) + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.3.2.2

Subtract $915$ from $2132$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 \cdot 1217 + 55 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.4

Evaluate $| \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |$ .

Tap for more steps...

Step 5.4.1

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

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 \cdot 1217 + 55 (82 \cdot 61 - 15 \cdot 172)) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.4.2

Simplify the determinant.

Tap for more steps...

Step 5.4.2.1

Simplify each term.

Tap for more steps...

Step 5.4.2.1.1

Multiply $82$ by $61$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 \cdot 1217 + 55 (5002 - 15 \cdot 172)) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.4.2.1.2

Multiply $- 15$ by $172$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 \cdot 1217 + 55 (5002 - 2580)) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.4.2.2

Subtract $2580$ from $5002$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (39 \cdot 751 - 131 \cdot 1217 + 55 \cdot 2422) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.5

Simplify the determinant.

Tap for more steps...

Step 5.5.1

Simplify each term.

Tap for more steps...

Step 5.5.1.1

Multiply $39$ by $751$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (29289 - 131 \cdot 1217 + 55 \cdot 2422) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.5.1.2

Multiply $- 131$ by $1217$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (29289 - 159427 + 55 \cdot 2422) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.5.1.3

Multiply $55$ by $2422$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (29289 - 159427 + 133210) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.5.2

Subtract $159427$ from $29289$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 (- 130138 + 133210) - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 5.5.3

Add $- 130138$ and $133210$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 | \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 39 & 131 & 160 \\ 82 & 172 & 185 \\ 15 & 61 & 77 \end{array} |$ .

Tap for more steps...

Step 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.

Tap for more steps...

Step 6.1.1

Consider the corresponding sign chart.

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

Step 6.1.2

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

Step 6.1.3

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

$| \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |$

Step 6.1.4

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

$39 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |$

Step 6.1.5

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

$| \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |$

Step 6.1.6

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

$- 131 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |$

Step 6.1.7

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

$| \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |$

Step 6.1.8

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

$160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |$

Step 6.1.9

Add the terms together.

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 | \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} | - 131 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} | + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.2

Evaluate $| \begin{array}{c} 172 & 185 \\ 61 & 77 \end{array} |$ .

Tap for more steps...

Step 6.2.1

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

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 (172 \cdot 77 - 61 \cdot 185) - 131 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} | + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.2.2

Simplify the determinant.

Tap for more steps...

Step 6.2.2.1

Simplify each term.

Tap for more steps...

Step 6.2.2.1.1

Multiply $172$ by $77$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 (13244 - 61 \cdot 185) - 131 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} | + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.2.2.1.2

Multiply $- 61$ by $185$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 (13244 - 11285) - 131 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} | + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.2.2.2

Subtract $11285$ from $13244$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 | \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} | + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.3

Evaluate $| \begin{array}{c} 82 & 185 \\ 15 & 77 \end{array} |$ .

Tap for more steps...

Step 6.3.1

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

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 (82 \cdot 77 - 15 \cdot 185) + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.3.2

Simplify the determinant.

Tap for more steps...

Step 6.3.2.1

Simplify each term.

Tap for more steps...

Step 6.3.2.1.1

Multiply $82$ by $77$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 (6314 - 15 \cdot 185) + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.3.2.1.2

Multiply $- 15$ by $185$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 (6314 - 2775) + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.3.2.2

Subtract $2775$ from $6314$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 \cdot 3539 + 160 | \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |)$

Step 6.4

Evaluate $| \begin{array}{c} 82 & 172 \\ 15 & 61 \end{array} |$ .

Tap for more steps...

Step 6.4.1

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

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 \cdot 3539 + 160 (82 \cdot 61 - 15 \cdot 172))$

Step 6.4.2

Simplify the determinant.

Tap for more steps...

Step 6.4.2.1

Simplify each term.

Tap for more steps...

Step 6.4.2.1.1

Multiply $82$ by $61$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 \cdot 3539 + 160 (5002 - 15 \cdot 172))$

Step 6.4.2.1.2

Multiply $- 15$ by $172$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 \cdot 3539 + 160 (5002 - 2580))$

Step 6.4.2.2

Subtract $2580$ from $5002$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (39 \cdot 1959 - 131 \cdot 3539 + 160 \cdot 2422)$

Step 6.5

Simplify the determinant.

Tap for more steps...

Step 6.5.1

Simplify each term.

Tap for more steps...

Step 6.5.1.1

Multiply $39$ by $1959$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (76401 - 131 \cdot 3539 + 160 \cdot 2422)$

Step 6.5.1.2

Multiply $- 131$ by $3539$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (76401 - 463609 + 160 \cdot 2422)$

Step 6.5.1.3

Multiply $160$ by $2422$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (76401 - 463609 + 387520)$

Step 6.5.2

Subtract $463609$ from $76401$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 (- 387208 + 387520)$

Step 6.5.3

Add $- 387208$ and $387520$ .

$39 \cdot 2388 - 131 \cdot 4332 + 160 \cdot 3072 - 55 \cdot 312$

Step 7

Simplify the determinant.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply $39$ by $2388$ .

$93132 - 131 \cdot 4332 + 160 \cdot 3072 - 55 \cdot 312$

Step 7.1.2

Multiply $- 131$ by $4332$ .

$93132 - 567492 + 160 \cdot 3072 - 55 \cdot 312$

Step 7.1.3

Multiply $160$ by $3072$ .

$93132 - 567492 + 491520 - 55 \cdot 312$

Step 7.1.4

Multiply $- 55$ by $312$ .

$93132 - 567492 + 491520 - 17160$

Step 7.2

Subtract $567492$ from $93132$ .

$- 474360 + 491520 - 17160$

Step 7.3

Add $- 474360$ and $491520$ .

$17160 - 17160$

Step 7.4

Subtract $17160$ from $17160$ .

$0$