Linear Algebra Examples

$[\begin{array}{c} - 2 & - 3 & 5 & - 7 \\ 3 & 2 & - 2 & 5 \\ 1 & - 2 & 4 & - 2 \\ 2 & 5 & - 7 & 5 \end{array}]$

Step 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 1.1

Consider the corresponding sign chart.

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

Step 1.2

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

Step 1.3

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

$| \begin{array}{c} 2 & - 2 & 5 \\ - 2 & 4 & - 2 \\ 5 & - 7 & 5 \end{array} |$

Step 1.4

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

$- 2 | \begin{array}{c} 2 & - 2 & 5 \\ - 2 & 4 & - 2 \\ 5 & - 7 & 5 \end{array} |$

Step 1.5

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

$| \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} |$

Step 1.6

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

$3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} |$

Step 1.7

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

$| \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} |$

Step 1.8

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

$5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} |$

Step 1.9

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

$| \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 1.10

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

$7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 1.11

Add the terms together.

$- 2 | \begin{array}{c} 2 & - 2 & 5 \\ - 2 & 4 & - 2 \\ 5 & - 7 & 5 \end{array} | + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 2 & - 2 & 5 \\ - 2 & 4 & - 2 \\ 5 & - 7 & 5 \end{array} |$ .

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Step 2.1

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

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Step 2.1.1

Consider the corresponding sign chart.

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

Step 2.1.2

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

Step 2.1.3

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

$| \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} |$

Step 2.1.4

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

$2 | \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} |$

Step 2.1.5

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

$| \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} |$

Step 2.1.6

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

$2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} |$

Step 2.1.7

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

$| \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |$

Step 2.1.8

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

$5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |$

Step 2.1.9

Add the terms together.

$- 2 (2 | \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} | + 2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.2

Evaluate $| \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} |$ .

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Step 2.2.1

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

$- 2 (2 (4 \cdot 5 - (- 7 \cdot - 2)) + 2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.2.2

Simplify the determinant.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $4$ by $5$ .

$- 2 (2 (20 - (- 7 \cdot - 2)) + 2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.2.2.1.2

Multiply $- (- 7 \cdot - 2)$ .

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Step 2.2.2.1.2.1

Multiply $- 7$ by $- 2$ .

$- 2 (2 (20 - 1 \cdot 14) + 2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.2.2.1.2.2

Multiply $- 1$ by $14$ .

$- 2 (2 (20 - 14) + 2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.2.2.2

Subtract $14$ from $20$ .

$- 2 (2 \cdot 6 + 2 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.3

Evaluate $| \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} |$ .

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Step 2.3.1

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

$- 2 (2 \cdot 6 + 2 (- 2 \cdot 5 - 5 \cdot - 2) + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.3.2

Simplify the determinant.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $- 2$ by $5$ .

$- 2 (2 \cdot 6 + 2 (- 10 - 5 \cdot - 2) + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.3.2.1.2

Multiply $- 5$ by $- 2$ .

$- 2 (2 \cdot 6 + 2 (- 10 + 10) + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.3.2.2

Add $- 10$ and $10$ .

$- 2 (2 \cdot 6 + 2 \cdot 0 + 5 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.4

Evaluate $| \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |$ .

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Step 2.4.1

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

$- 2 (2 \cdot 6 + 2 \cdot 0 + 5 (- 2 \cdot - 7 - 5 \cdot 4)) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.4.2

Simplify the determinant.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $- 2$ by $- 7$ .

$- 2 (2 \cdot 6 + 2 \cdot 0 + 5 (14 - 5 \cdot 4)) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.4.2.1.2

Multiply $- 5$ by $4$ .

$- 2 (2 \cdot 6 + 2 \cdot 0 + 5 (14 - 20)) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.4.2.2

Subtract $20$ from $14$ .

$- 2 (2 \cdot 6 + 2 \cdot 0 + 5 \cdot - 6) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.5

Simplify the determinant.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $2$ by $6$ .

$- 2 (12 + 2 \cdot 0 + 5 \cdot - 6) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.5.1.2

Multiply $2$ by $0$ .

$- 2 (12 + 0 + 5 \cdot - 6) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.5.1.3

Multiply $5$ by $- 6$ .

$- 2 (12 + 0 - 30) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.5.2

Add $12$ and $0$ .

$- 2 (12 - 30) + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 2.5.3

Subtract $30$ from $12$ .

$- 2 \cdot - 18 + 3 | \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} | + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 3 & - 2 & 5 \\ 1 & 4 & - 2 \\ 2 & - 7 & 5 \end{array} |$ .

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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.

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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.

$| \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} |$

Step 3.1.4

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

$3 | \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} |$

Step 3.1.5

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

$| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 3.1.6

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

$2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 3.1.7

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

$| \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |$

Step 3.1.8

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

$5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |$

Step 3.1.9

Add the terms together.

$- 2 \cdot - 18 + 3 (3 | \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} | + 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.2

Evaluate $| \begin{array}{c} 4 & - 2 \\ - 7 & 5 \end{array} |$ .

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Step 3.2.1

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

$- 2 \cdot - 18 + 3 (3 (4 \cdot 5 - (- 7 \cdot - 2)) + 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.2.2

Simplify the determinant.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $4$ by $5$ .

$- 2 \cdot - 18 + 3 (3 (20 - (- 7 \cdot - 2)) + 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.2.2.1.2

Multiply $- (- 7 \cdot - 2)$ .

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Step 3.2.2.1.2.1

Multiply $- 7$ by $- 2$ .

$- 2 \cdot - 18 + 3 (3 (20 - 1 \cdot 14) + 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.2.2.1.2.2

Multiply $- 1$ by $14$ .

$- 2 \cdot - 18 + 3 (3 (20 - 14) + 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.2.2.2

Subtract $14$ from $20$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.3

Evaluate $| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$ .

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Step 3.3.1

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

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 (1 \cdot 5 - 2 \cdot - 2) + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.3.2

Simplify the determinant.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $5$ by $1$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 (5 - 2 \cdot - 2) + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.3.2.1.2

Multiply $- 2$ by $- 2$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 (5 + 4) + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.3.2.2

Add $5$ and $4$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 \cdot 9 + 5 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.4

Evaluate $| \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |$ .

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Step 3.4.1

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

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 \cdot 9 + 5 (1 \cdot - 7 - 2 \cdot 4)) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.4.2

Simplify the determinant.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply $- 7$ by $1$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 \cdot 9 + 5 (- 7 - 2 \cdot 4)) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.4.2.1.2

Multiply $- 2$ by $4$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 \cdot 9 + 5 (- 7 - 8)) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.4.2.2

Subtract $8$ from $- 7$ .

$- 2 \cdot - 18 + 3 (3 \cdot 6 + 2 \cdot 9 + 5 \cdot - 15) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.5

Simplify the determinant.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $3$ by $6$ .

$- 2 \cdot - 18 + 3 (18 + 2 \cdot 9 + 5 \cdot - 15) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.5.1.2

Multiply $2$ by $9$ .

$- 2 \cdot - 18 + 3 (18 + 18 + 5 \cdot - 15) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.5.1.3

Multiply $5$ by $- 15$ .

$- 2 \cdot - 18 + 3 (18 + 18 - 75) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.5.2

Add $18$ and $18$ .

$- 2 \cdot - 18 + 3 (36 - 75) + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 3.5.3

Subtract $75$ from $36$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 | \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} | + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 3 & 2 & 5 \\ 1 & - 2 & - 2 \\ 2 & 5 & 5 \end{array} |$ .

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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.

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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.

$| \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} |$

Step 4.1.4

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

$3 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} |$

Step 4.1.5

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

$| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 4.1.6

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

$- 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 4.1.7

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

$| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 4.1.8

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

$5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 4.1.9

Add the terms together.

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 | \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} | - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.2

Evaluate $| \begin{array}{c} - 2 & - 2 \\ 5 & 5 \end{array} |$ .

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Step 4.2.1

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

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 (- 2 \cdot 5 - 5 \cdot - 2) - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.2.2

Simplify the determinant.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $- 2$ by $5$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 (- 10 - 5 \cdot - 2) - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.2.2.1.2

Multiply $- 5$ by $- 2$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 (- 10 + 10) - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.2.2.2

Add $- 10$ and $10$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} | + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.3

Evaluate $| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$ .

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Step 4.3.1

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

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 (1 \cdot 5 - 2 \cdot - 2) + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.3.2

Simplify the determinant.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $5$ by $1$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 (5 - 2 \cdot - 2) + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.3.2.1.2

Multiply $- 2$ by $- 2$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 (5 + 4) + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.3.2.2

Add $5$ and $4$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 \cdot 9 + 5 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.4

Evaluate $| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$ .

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Step 4.4.1

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

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 \cdot 9 + 5 (1 \cdot 5 - 2 \cdot - 2)) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.4.2

Simplify the determinant.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Multiply $5$ by $1$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 \cdot 9 + 5 (5 - 2 \cdot - 2)) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.4.2.1.2

Multiply $- 2$ by $- 2$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 \cdot 9 + 5 (5 + 4)) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.4.2.2

Add $5$ and $4$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (3 \cdot 0 - 2 \cdot 9 + 5 \cdot 9) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.5

Simplify the determinant.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $3$ by $0$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (0 - 2 \cdot 9 + 5 \cdot 9) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.5.1.2

Multiply $- 2$ by $9$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (0 - 18 + 5 \cdot 9) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.5.1.3

Multiply $5$ by $9$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (0 - 18 + 45) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.5.2

Subtract $18$ from $0$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 (- 18 + 45) + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 4.5.3

Add $- 18$ and $45$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 | \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 3 & 2 & - 2 \\ 1 & - 2 & 4 \\ 2 & 5 & - 7 \end{array} |$ .

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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.

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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.

$| \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |$

Step 5.1.4

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

$3 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |$

Step 5.1.6

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

$- 2 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 5.1.8

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

$- 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$

Step 5.1.9

Add the terms together.

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 | \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} | - 2 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} | - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.2

Evaluate $| \begin{array}{c} - 2 & 4 \\ 5 & - 7 \end{array} |$ .

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Step 5.2.1

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

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 (- 2 \cdot - 7 - 5 \cdot 4) - 2 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} | - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.2.2

Simplify the determinant.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $- 2$ by $- 7$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 (14 - 5 \cdot 4) - 2 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} | - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.2.2.1.2

Multiply $- 5$ by $4$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 (14 - 20) - 2 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} | - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.2.2.2

Subtract $20$ from $14$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 | \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} | - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.3

Evaluate $| \begin{array}{c} 1 & 4 \\ 2 & - 7 \end{array} |$ .

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Step 5.3.1

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

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 (1 \cdot - 7 - 2 \cdot 4) - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.3.2

Simplify the determinant.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $- 7$ by $1$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 (- 7 - 2 \cdot 4) - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.3.2.1.2

Multiply $- 2$ by $4$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 (- 7 - 8) - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.3.2.2

Subtract $8$ from $- 7$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 \cdot - 15 - 2 | \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |)$

Step 5.4

Evaluate $| \begin{array}{c} 1 & - 2 \\ 2 & 5 \end{array} |$ .

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Step 5.4.1

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

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 \cdot - 15 - 2 (1 \cdot 5 - 2 \cdot - 2))$

Step 5.4.2

Simplify the determinant.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Multiply $5$ by $1$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 \cdot - 15 - 2 (5 - 2 \cdot - 2))$

Step 5.4.2.1.2

Multiply $- 2$ by $- 2$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 \cdot - 15 - 2 (5 + 4))$

Step 5.4.2.2

Add $5$ and $4$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (3 \cdot - 6 - 2 \cdot - 15 - 2 \cdot 9)$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $3$ by $- 6$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (- 18 - 2 \cdot - 15 - 2 \cdot 9)$

Step 5.5.1.2

Multiply $- 2$ by $- 15$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (- 18 + 30 - 2 \cdot 9)$

Step 5.5.1.3

Multiply $- 2$ by $9$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (- 18 + 30 - 18)$

Step 5.5.2

Add $- 18$ and $30$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 (12 - 18)$

Step 5.5.3

Subtract $18$ from $12$ .

$- 2 \cdot - 18 + 3 \cdot - 39 + 5 \cdot 27 + 7 \cdot - 6$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $- 2$ by $- 18$ .

$36 + 3 \cdot - 39 + 5 \cdot 27 + 7 \cdot - 6$

Step 6.1.2

Multiply $3$ by $- 39$ .

$36 - 117 + 5 \cdot 27 + 7 \cdot - 6$

Step 6.1.3

Multiply $5$ by $27$ .

$36 - 117 + 135 + 7 \cdot - 6$

Step 6.1.4

Multiply $7$ by $- 6$ .

$36 - 117 + 135 - 42$

Step 6.2

Subtract $117$ from $36$ .

$- 81 + 135 - 42$

Step 6.3

Add $- 81$ and $135$ .

$54 - 42$

Step 6.4

Subtract $42$ from $54$ .

$12$