Linear Algebra Examples

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

Refer to the corresponding sign matrix below.

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

Use the sign matrix and the given matrix, $A$ , to find the cofactor of each element.

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The determinant of $[\begin{array}{c} 5 & 6 \\ 2 & 3 \end{array}]$ is $3$ .

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

$A_{11} = (5) (3) - 2 \cdot 6$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $3$ .

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

Multiply $- 2$ by $6$ .

$A_{11} = 15 - 12$

Subtract $12$ from $15$ .

$A_{11} = 3$

The determinant of $- [\begin{array}{c} 4 & 6 \\ 1 & 3 \end{array}]$ is $- 6$ .

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

$A_{12} = - ((4) (3) - 1 \cdot 6)$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $3$ .

$A_{12} = - (12 - 1 \cdot 6)$

Multiply $- 1$ by $6$ .

$A_{12} = - (12 - 6)$

Simplify the expression.

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Subtract $6$ from $12$ .

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

Multiply $- 1$ by $6$ .

$A_{12} = - 6$

The determinant of $[\begin{array}{c} 4 & 5 \\ 1 & 2 \end{array}]$ is $3$ .

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

$A_{13} = (4) (2) - 1 \cdot 5$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $2$ .

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

Multiply $- 1$ by $5$ .

$A_{13} = 8 - 5$

Subtract $5$ from $8$ .

$A_{13} = 3$

The determinant of $- [\begin{array}{c} 8 & 7 \\ 2 & 3 \end{array}]$ is $- 10$ .

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

$A_{21} = - ((8) (3) - 2 \cdot 7)$

Simplify the determinant.

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Simplify each term.

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Multiply $8$ by $3$ .

$A_{21} = - (24 - 2 \cdot 7)$

Multiply $- 2$ by $7$ .

$A_{21} = - (24 - 14)$

Simplify the expression.

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Subtract $14$ from $24$ .

$A_{21} = - 1 \cdot 10$

Multiply $- 1$ by $10$ .

$A_{21} = - 10$

The determinant of $[\begin{array}{c} 9 & 7 \\ 1 & 3 \end{array}]$ is $20$ .

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

$A_{22} = (9) (3) - 1 \cdot 7$

Simplify the determinant.

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Simplify each term.

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Multiply $9$ by $3$ .

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

Multiply $- 1$ by $7$ .

$A_{22} = 27 - 7$

Subtract $7$ from $27$ .

$A_{22} = 20$

The determinant of $- [\begin{array}{c} 9 & 8 \\ 1 & 2 \end{array}]$ is $- 10$ .

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

$A_{23} = - ((9) (2) - 1 \cdot 8)$

Simplify the determinant.

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Simplify each term.

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Multiply $9$ by $2$ .

$A_{23} = - (18 - 1 \cdot 8)$

Multiply $- 1$ by $8$ .

$A_{23} = - (18 - 8)$

Simplify the expression.

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Subtract $8$ from $18$ .

$A_{23} = - 1 \cdot 10$

Multiply $- 1$ by $10$ .

$A_{23} = - 10$

The determinant of $[\begin{array}{c} 8 & 7 \\ 5 & 6 \end{array}]$ is $13$ .

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

$A_{31} = (8) (6) - 5 \cdot 7$

Simplify the determinant.

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Simplify each term.

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Multiply $8$ by $6$ .

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

Multiply $- 5$ by $7$ .

$A_{31} = 48 - 35$

Subtract $35$ from $48$ .

$A_{31} = 13$

The determinant of $- [\begin{array}{c} 9 & 7 \\ 4 & 6 \end{array}]$ is $- 26$ .

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

$A_{32} = - ((9) (6) - 4 \cdot 7)$

Simplify the determinant.

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Simplify each term.

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Multiply $9$ by $6$ .

$A_{32} = - (54 - 4 \cdot 7)$

Multiply $- 4$ by $7$ .

$A_{32} = - (54 - 28)$

Simplify the expression.

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Subtract $28$ from $54$ .

$A_{32} = - 1 \cdot 26$

Multiply $- 1$ by $26$ .

$A_{32} = - 26$

The determinant of $[\begin{array}{c} 9 & 8 \\ 4 & 5 \end{array}]$ is $13$ .

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

$A_{33} = (9) (5) - 4 \cdot 8$

Simplify the determinant.

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Simplify each term.

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Multiply $9$ by $5$ .

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

Multiply $- 4$ by $8$ .

$A_{33} = 45 - 32$

Subtract $32$ from $45$ .

$A_{33} = 13$

The cofactor matrix is a matrix with cofactors as the elements.

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

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