Examples

$[\begin{array}{c} 3 & 2 & - 1 \\ 1 & 6 & 3 \\ 2 & - 4 & 0 \end{array}]$

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.

$| \begin{array}{c} 6 & 3 \\ - 4 & 0 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

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

$a_{11} = 6 \cdot 0 - (- 4 \cdot 3)$

Step 2.1.2.2

Simplify the determinant.

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Step 2.1.2.2.1

Simplify each term.

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Step 2.1.2.2.1.1

Multiply $6$ by $0$ .

$a_{11} = 0 - (- 4 \cdot 3)$

Step 2.1.2.2.1.2

Multiply $- (- 4 \cdot 3)$ .

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Step 2.1.2.2.1.2.1

Multiply $- 4$ by $3$ .

$a_{11} = 0 - - 12$

Step 2.1.2.2.1.2.2

Multiply $- 1$ by $- 12$ .

$a_{11} = 0 + 12$

Step 2.1.2.2.2

Add $0$ and $12$ .

$a_{11} = 12$

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.

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

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

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

$a_{12} = 1 \cdot 0 - 2 \cdot 3$

Step 2.2.2.2

Simplify the determinant.

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Step 2.2.2.2.1

Simplify each term.

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Step 2.2.2.2.1.1

Multiply $0$ by $1$ .

$a_{12} = 0 - 2 \cdot 3$

Step 2.2.2.2.1.2

Multiply $- 2$ by $3$ .

$a_{12} = 0 - 6$

Step 2.2.2.2.2

Subtract $6$ from $0$ .

$a_{12} = - 6$

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.

$| \begin{array}{c} 1 & 6 \\ 2 & - 4 \end{array} |$

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

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

$a_{13} = 1 \cdot - 4 - 2 \cdot 6$

Step 2.3.2.2

Simplify the determinant.

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Step 2.3.2.2.1

Simplify each term.

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Step 2.3.2.2.1.1

Multiply $- 4$ by $1$ .

$a_{13} = - 4 - 2 \cdot 6$

Step 2.3.2.2.1.2

Multiply $- 2$ by $6$ .

$a_{13} = - 4 - 12$

Step 2.3.2.2.2

Subtract $12$ from $- 4$ .

$a_{13} = - 16$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

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

$a_{21} = 2 \cdot 0 - (- 4 \cdot - 1)$

Step 2.4.2.2

Simplify the determinant.

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Step 2.4.2.2.1

Simplify each term.

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Step 2.4.2.2.1.1

Multiply $2$ by $0$ .

$a_{21} = 0 - (- 4 \cdot - 1)$

Step 2.4.2.2.1.2

Multiply $- (- 4 \cdot - 1)$ .

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Step 2.4.2.2.1.2.1

Multiply $- 4$ by $- 1$ .

$a_{21} = 0 - 1 \cdot 4$

Step 2.4.2.2.1.2.2

Multiply $- 1$ by $4$ .

$a_{21} = 0 - 4$

Step 2.4.2.2.2

Subtract $4$ from $0$ .

$a_{21} = - 4$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

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

$a_{22} = 3 \cdot 0 - 2 \cdot - 1$

Step 2.5.2.2

Simplify the determinant.

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Step 2.5.2.2.1

Simplify each term.

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Step 2.5.2.2.1.1

Multiply $3$ by $0$ .

$a_{22} = 0 - 2 \cdot - 1$

Step 2.5.2.2.1.2

Multiply $- 2$ by $- 1$ .

$a_{22} = 0 + 2$

Step 2.5.2.2.2

Add $0$ and $2$ .

$a_{22} = 2$

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

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

$a_{23} = 3 \cdot - 4 - 2 \cdot 2$

Step 2.6.2.2

Simplify the determinant.

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Step 2.6.2.2.1

Simplify each term.

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Step 2.6.2.2.1.1

Multiply $3$ by $- 4$ .

$a_{23} = - 12 - 2 \cdot 2$

Step 2.6.2.2.1.2

Multiply $- 2$ by $2$ .

$a_{23} = - 12 - 4$

Step 2.6.2.2.2

Subtract $4$ from $- 12$ .

$a_{23} = - 16$

Step 2.7

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

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Step 2.7.1

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

$| \begin{array}{c} 2 & - 1 \\ 6 & 3 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

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

$a_{31} = 2 \cdot 3 - 6 \cdot - 1$

Step 2.7.2.2

Simplify the determinant.

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Step 2.7.2.2.1

Simplify each term.

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Step 2.7.2.2.1.1

Multiply $2$ by $3$ .

$a_{31} = 6 - 6 \cdot - 1$

Step 2.7.2.2.1.2

Multiply $- 6$ by $- 1$ .

$a_{31} = 6 + 6$

Step 2.7.2.2.2

Add $6$ and $6$ .

$a_{31} = 12$

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

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

$a_{32} = 3 \cdot 3 - 1 \cdot - 1$

Step 2.8.2.2

Simplify the determinant.

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Step 2.8.2.2.1

Simplify each term.

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Step 2.8.2.2.1.1

Multiply $3$ by $3$ .

$a_{32} = 9 - 1 \cdot - 1$

Step 2.8.2.2.1.2

Multiply $- 1$ by $- 1$ .

$a_{32} = 9 + 1$

Step 2.8.2.2.2

Add $9$ and $1$ .

$a_{32} = 10$

Step 2.9

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

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Step 2.9.1

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

$| \begin{array}{c} 3 & 2 \\ 1 & 6 \end{array} |$

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

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

$a_{33} = 3 \cdot 6 - 1 \cdot 2$

Step 2.9.2.2

Simplify the determinant.

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Step 2.9.2.2.1

Simplify each term.

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Step 2.9.2.2.1.1

Multiply $3$ by $6$ .

$a_{33} = 18 - 1 \cdot 2$

Step 2.9.2.2.1.2

Multiply $- 1$ by $2$ .

$a_{33} = 18 - 2$

Step 2.9.2.2.2

Subtract $2$ from $18$ .

$a_{33} = 16$

Step 2.10

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

$[\begin{array}{c} 12 & 6 & - 16 \\ 4 & 2 & 16 \\ 12 & - 10 & 16 \end{array}]$

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