Examples

⎡ ⎢ ⎣ \begin{matrix} 9 & 8 & 7 4 & 5 & 6 1 & 2 & 3 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 9 & 8 & 7 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{array}]$

Step 1

Consider the corresponding sign chart.

⎡ ⎢ ⎣ \begin{matrix} + & - & + - & + & - + & - & + \end{matrix} ⎤ ⎥ ⎦

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

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}

$a_{11}$ .

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Step 2.1.1

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 5 & 6 2 & 3 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 5 & 6 \\ 2 & 3 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{11} = 5 \cdot 3 - 2 \cdot 6

$a_{11} = 5 \cdot 3 - 2 \cdot 6$

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

5

$5$ by

3

$3$ .

a_{11} = 15 - 2 \cdot 6

$a_{11} = 15 - 2 \cdot 6$

Step 2.1.2.2.1.2

Multiply

- 2

$- 2$ by

6

$6$ .

a_{11} = 15 - 12

$a_{11} = 15 - 12$

a_{11} = 15 - 12

$a_{11} = 15 - 12$

Step 2.1.2.2.2

Subtract

12

$12$ from

15

$15$ .

a_{11} = 3

$a_{11} = 3$

a_{11} = 3

$a_{11} = 3$

a_{11} = 3

$a_{11} = 3$

a_{11} = 3

$a_{11} = 3$

Step 2.2

Calculate the minor for element

a_{12}

$a_{12}$ .

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Step 2.2.1

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 4 & 6 1 & 3 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & 6 \\ 1 & 3 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{12} = 4 \cdot 3 - 1 \cdot 6

$a_{12} = 4 \cdot 3 - 1 \cdot 6$

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

4

$4$ by

3

$3$ .

a_{12} = 12 - 1 \cdot 6

$a_{12} = 12 - 1 \cdot 6$

Step 2.2.2.2.1.2

Multiply

- 1

$- 1$ by

6

$6$ .

a_{12} = 12 - 6

$a_{12} = 12 - 6$

a_{12} = 12 - 6

$a_{12} = 12 - 6$

Step 2.2.2.2.2

Subtract

6

$6$ from

12

$12$ .

a_{12} = 6

$a_{12} = 6$

a_{12} = 6

$a_{12} = 6$

a_{12} = 6

$a_{12} = 6$

a_{12} = 6

$a_{12} = 6$

Step 2.3

Calculate the minor for element

a_{13}

$a_{13}$ .

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Step 2.3.1

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 4 & 5 1 & 2 \end{matrix} ∣ ∣ ∣

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

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{13} = 4 \cdot 2 - 1 \cdot 5

$a_{13} = 4 \cdot 2 - 1 \cdot 5$

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

$4$ by

2

$2$ .

a_{13} = 8 - 1 \cdot 5

$a_{13} = 8 - 1 \cdot 5$

Step 2.3.2.2.1.2

Multiply

- 1

$- 1$ by

5

$5$ .

a_{13} = 8 - 5

$a_{13} = 8 - 5$

a_{13} = 8 - 5

$a_{13} = 8 - 5$

Step 2.3.2.2.2

Subtract

5

$5$ from

8

$8$ .

a_{13} = 3

$a_{13} = 3$

a_{13} = 3

$a_{13} = 3$

a_{13} = 3

$a_{13} = 3$

a_{13} = 3

$a_{13} = 3$

Step 2.4

Calculate the minor for element

a_{21}

$a_{21}$ .

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Step 2.4.1

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 8 & 7 2 & 3 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 8 & 7 \\ 2 & 3 \end{array} |$

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{21} = 8 \cdot 3 - 2 \cdot 7

$a_{21} = 8 \cdot 3 - 2 \cdot 7$

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

8

$8$ by

3

$3$ .

a_{21} = 24 - 2 \cdot 7

$a_{21} = 24 - 2 \cdot 7$

Step 2.4.2.2.1.2

Multiply

- 2

$- 2$ by

7

$7$ .

a_{21} = 24 - 14

$a_{21} = 24 - 14$

a_{21} = 24 - 14

$a_{21} = 24 - 14$

Step 2.4.2.2.2

Subtract

14

$14$ from

24

$24$ .

a_{21} = 10

$a_{21} = 10$

a_{21} = 10

$a_{21} = 10$

a_{21} = 10

$a_{21} = 10$

a_{21} = 10

$a_{21} = 10$

Step 2.5

Calculate the minor for element

a_{22}

$a_{22}$ .

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Step 2.5.1

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 9 & 7 1 & 3 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 9 & 7 \\ 1 & 3 \end{array} |$

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{22} = 9 \cdot 3 - 1 \cdot 7

$a_{22} = 9 \cdot 3 - 1 \cdot 7$

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

9

$9$ by

3

$3$ .

a_{22} = 27 - 1 \cdot 7

$a_{22} = 27 - 1 \cdot 7$

Step 2.5.2.2.1.2

Multiply

- 1

$- 1$ by

7

$7$ .

a_{22} = 27 - 7

$a_{22} = 27 - 7$

a_{22} = 27 - 7

$a_{22} = 27 - 7$

Step 2.5.2.2.2

Subtract

7

$7$ from

27

$27$ .

a_{22} = 20

$a_{22} = 20$

a_{22} = 20

$a_{22} = 20$

a_{22} = 20

$a_{22} = 20$

a_{22} = 20

$a_{22} = 20$

Step 2.6

Calculate the minor for element

a_{23}

$a_{23}$ .

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Step 2.6.1

The minor for

a_{23}

$a_{23}$ is the determinant with row

2

$2$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 9 & 8 1 & 2 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 9 & 8 \\ 1 & 2 \end{array} |$

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{23} = 9 \cdot 2 - 1 \cdot 8

$a_{23} = 9 \cdot 2 - 1 \cdot 8$

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

9

$9$ by

2

$2$ .

a_{23} = 18 - 1 \cdot 8

$a_{23} = 18 - 1 \cdot 8$

Step 2.6.2.2.1.2

Multiply

- 1

$- 1$ by

8

$8$ .

a_{23} = 18 - 8

$a_{23} = 18 - 8$

a_{23} = 18 - 8

$a_{23} = 18 - 8$

Step 2.6.2.2.2

Subtract

8

$8$ from

18

$18$ .

a_{23} = 10

$a_{23} = 10$

a_{23} = 10

$a_{23} = 10$

a_{23} = 10

$a_{23} = 10$

a_{23} = 10

$a_{23} = 10$

Step 2.7

Calculate the minor for element

a_{31}

$a_{31}$ .

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Step 2.7.1

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 8 & 7 5 & 6 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 8 & 7 \\ 5 & 6 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

a_{31} = 8 \cdot 6 - 5 \cdot 7

$a_{31} = 8 \cdot 6 - 5 \cdot 7$

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

8

$8$ by

6

$6$ .

a_{31} = 48 - 5 \cdot 7

$a_{31} = 48 - 5 \cdot 7$

Step 2.7.2.2.1.2

Multiply

- 5

$- 5$ by

7

$7$ .

a_{31} = 48 - 35

$a_{31} = 48 - 35$

a_{31} = 48 - 35

$a_{31} = 48 - 35$

Step 2.7.2.2.2

Subtract

35

$35$ from

48

$48$ .

a_{31} = 13

$a_{31} = 13$

a_{31} = 13

$a_{31} = 13$

a_{31} = 13

$a_{31} = 13$

a_{31} = 13

$a_{31} = 13$

Step 2.8

Calculate the minor for element

a_{32}

$a_{32}$ .

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Step 2.8.1

The minor for

a_{32}

$a_{32}$ is the determinant with row

3

$3$ and column

2

$2$ deleted.

$| \begin{array}{c} 9 & 7 \\ 4 & 6 \end{array} |$

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

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{32} = 9 \cdot 6 - 4 \cdot 7$

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

$9$ by

$6$ .

$a_{32} = 54 - 4 \cdot 7$

Step 2.8.2.2.1.2

Multiply

$- 4$ by

$7$ .

$a_{32} = 54 - 28$

Step 2.8.2.2.2

Subtract

$28$ from

$54$ .

$a_{32} = 26$

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.

$| \begin{array}{c} 9 & 8 \\ 4 & 5 \end{array} |$

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

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{33} = 9 \cdot 5 - 4 \cdot 8$

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

$9$ by

$5$ .

$a_{33} = 45 - 4 \cdot 8$

Step 2.9.2.2.1.2

Multiply

$- 4$ by

$8$ .

$a_{33} = 45 - 32$

Step 2.9.2.2.2

Subtract

$32$ from

$45$ .

$a_{33} = 13$

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.

$[\begin{array}{c} 3 & - 6 & 3 \\ - 10 & 20 & - 10 \\ 13 & - 26 & 13 \end{array}]$

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