Examples

$[\begin{array}{c} 0 & 3 & 3 \\ 1 & 4 & 0 \\ 1 & 1 & 2 \end{array}]$

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.

$| \begin{array}{c} 4 & 0 \\ 1 & 2 \end{array} |$

Step 1.4

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

$0 | \begin{array}{c} 4 & 0 \\ 1 & 2 \end{array} |$

Step 1.5

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

$| \begin{array}{c} 1 & 0 \\ 1 & 2 \end{array} |$

Step 1.6

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

$- 3 | \begin{array}{c} 1 & 0 \\ 1 & 2 \end{array} |$

Step 1.7

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

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

Step 1.8

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

$3 | \begin{array}{c} 1 & 4 \\ 1 & 1 \end{array} |$

Step 1.9

Add the terms together.

$0 | \begin{array}{c} 4 & 0 \\ 1 & 2 \end{array} | - 3 | \begin{array}{c} 1 & 0 \\ 1 & 2 \end{array} | + 3 | \begin{array}{c} 1 & 4 \\ 1 & 1 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 4 & 0 \\ 1 & 2 \end{array} |$ .

$0 - 3 | \begin{array}{c} 1 & 0 \\ 1 & 2 \end{array} | + 3 | \begin{array}{c} 1 & 4 \\ 1 & 1 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & 0 \\ 1 & 2 \end{array} |$ .

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Step 3.1

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

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

Step 3.2

Simplify the determinant.

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Step 3.2.1

Multiply $2$ by $1$ .

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

Step 3.2.2

Subtract $0$ from $2$ .

$0 - 3 \cdot 2 + 3 | \begin{array}{c} 1 & 4 \\ 1 & 1 \end{array} |$

Step 4

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

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Step 4.1

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

$0 - 3 \cdot 2 + 3 (1 \cdot 1 - 1 \cdot 4)$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $1$ by $1$ .

$0 - 3 \cdot 2 + 3 (1 - 1 \cdot 4)$

Step 4.2.1.2

Multiply $- 1$ by $4$ .

$0 - 3 \cdot 2 + 3 (1 - 4)$

Step 4.2.2

Subtract $4$ from $1$ .

$0 - 3 \cdot 2 + 3 \cdot - 3$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $- 3$ by $2$ .

$0 - 6 + 3 \cdot - 3$

Step 5.1.2

Multiply $3$ by $- 3$ .

$0 - 6 - 9$

Step 5.2

Subtract $6$ from $0$ .

$- 6 - 9$

Step 5.3

Subtract $9$ from $- 6$ .

$- 15$

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