Algebra Examples

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

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.

$| \begin{array}{c} 5 & 6 \\ 8 & 9 \end{array} |$

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

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

$a_{11} = 5 \cdot 9 - 8 \cdot 6$

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

$5$ by

$9$ .

$a_{11} = 45 - 8 \cdot 6$

Step 2.1.2.2.1.2

Multiply

$- 8$ by

$6$ .

$a_{11} = 45 - 48$

Step 2.1.2.2.2

Subtract

$48$ from

$45$ .

$a_{11} = - 3$

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.

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

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

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

$a_{12} = 4 \cdot 9 - 7 \cdot 6$

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

$4$ by

$9$ .

$a_{12} = 36 - 7 \cdot 6$

Step 2.2.2.2.1.2

Multiply

$- 7$ by

$6$ .

$a_{12} = 36 - 42$

Step 2.2.2.2.2

Subtract

$42$ from

$36$ .

$a_{12} = - 6$

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.

$| \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

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

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

$a_{13} = 4 \cdot 8 - 7 \cdot 5$

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

$4$ by

$8$ .

$a_{13} = 32 - 7 \cdot 5$

Step 2.3.2.2.1.2

Multiply

$- 7$ by

$5$ .

$a_{13} = 32 - 35$

Step 2.3.2.2.2

Subtract

$35$ from

$32$ .

$a_{13} = - 3$

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.

$| \begin{array}{c} 2 & 3 \\ 8 & 9 \end{array} |$

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

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

$a_{21} = 2 \cdot 9 - 8 \cdot 3$

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

$9$ .

$a_{21} = 18 - 8 \cdot 3$

Step 2.4.2.2.1.2

Multiply

$- 8$ by

$3$ .

$a_{21} = 18 - 24$

Step 2.4.2.2.2

Subtract

$24$ from

$18$ .

$a_{21} = - 6$

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.

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

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

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

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

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

$9$ by

$1$ .

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

Step 2.5.2.2.1.2

Multiply

$- 7$ by

$3$ .

$a_{22} = 9 - 21$

Step 2.5.2.2.2

Subtract

$21$ from

$9$ .

$a_{22} = - 12$

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.

$| \begin{array}{c} 1 & 2 \\ 7 & 8 \end{array} |$

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

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

$a_{23} = 1 \cdot 8 - 7 \cdot 2$

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

$8$ by

$1$ .

$a_{23} = 8 - 7 \cdot 2$

Step 2.6.2.2.1.2

Multiply

$- 7$ by

$2$ .

$a_{23} = 8 - 14$

Step 2.6.2.2.2

Subtract

$14$ from

$8$ .

$a_{23} = - 6$

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.

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

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

The determinant of a

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

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

$a_{31} = 2 \cdot 6 - 5 \cdot 3$

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

$6$ .

$a_{31} = 12 - 5 \cdot 3$

Step 2.7.2.2.1.2

Multiply

$- 5$ by

$3$ .

$a_{31} = 12 - 15$

Step 2.7.2.2.2

Subtract

$15$ from

$12$ .

$a_{31} = - 3$

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.

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

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

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

$a_{32} = 1 \cdot 6 - 4 \cdot 3$

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

$6$ by

$1$ .

$a_{32} = 6 - 4 \cdot 3$

Step 2.8.2.2.1.2

Multiply

$- 4$ by

$3$ .

$a_{32} = 6 - 12$

Step 2.8.2.2.2

Subtract

$12$ from

$6$ .

$a_{32} = - 6$

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.

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

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

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

$a_{33} = 1 \cdot 5 - 4 \cdot 2$

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

$5$ by

$1$ .

$a_{33} = 5 - 4 \cdot 2$

Step 2.9.2.2.1.2

Multiply

$- 4$ by

$2$ .

$a_{33} = 5 - 8$

Step 2.9.2.2.2

Subtract

$8$ from

$5$ .

$a_{33} = - 3$

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.

$[\begin{array}{c} - 3 & 6 & - 3 \\ 6 & - 12 & 6 \\ - 3 & 6 & - 3 \end{array}]$

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