Linear Algebra Examples

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

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 $[\begin{array}{c} 4 & 4 \\ 2 & 3 \end{array}]$ is $4$ .

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

$A_{11} = (4) (3) - 2 \cdot 4$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $3$ .

$A_{11} = 12 - 2 \cdot 4$

Multiply $- 2$ by $4$ .

$A_{11} = 12 - 8$

Subtract $8$ from $12$ .

$A_{11} = 4$

The determinant of $- [\begin{array}{c} 4 & 4 \\ 1 & 3 \end{array}]$ is $- 8$ .

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

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

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $3$ .

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

Multiply $- 1$ by $4$ .

$A_{12} = - (12 - 4)$

Simplify the expression.

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Subtract $4$ from $12$ .

$A_{12} = - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$A_{12} = - 8$

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

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

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

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $2$ .

$A_{13} = 8 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$A_{13} = 8 - 4$

Subtract $4$ from $8$ .

$A_{13} = 4$

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

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

$A_{21} = - ((2) (3) - 2 \cdot 1)$

Simplify the determinant.

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Simplify each term.

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Multiply $2$ by $3$ .

$A_{21} = - (6 - 2 \cdot 1)$

Multiply $- 2$ by $1$ .

$A_{21} = - (6 - 2)$

Simplify the expression.

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Subtract $2$ from $6$ .

$A_{21} = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$A_{21} = - 4$

The determinant of $[\begin{array}{c} 3 & 1 \\ 1 & 3 \end{array}]$ is $8$ .

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

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

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $3$ .

$A_{22} = 9 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$A_{22} = 9 - 1$

Subtract $1$ from $9$ .

$A_{22} = 8$

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

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

$A_{23} = - ((3) (2) - 1 \cdot 2)$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $2$ .

$A_{23} = - (6 - 1 \cdot 2)$

Multiply $- 1$ by $2$ .

$A_{23} = - (6 - 2)$

Simplify the expression.

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Subtract $2$ from $6$ .

$A_{23} = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$A_{23} = - 4$

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

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

$A_{31} = (2) (4) - 4 \cdot 1$

Simplify the determinant.

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Simplify each term.

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Multiply $2$ by $4$ .

$A_{31} = 8 - 4 \cdot 1$

Multiply $- 4$ by $1$ .

$A_{31} = 8 - 4$

Subtract $4$ from $8$ .

$A_{31} = 4$

The determinant of $- [\begin{array}{c} 3 & 1 \\ 4 & 4 \end{array}]$ is $- 8$ .

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

$A_{32} = - ((3) (4) - 4 \cdot 1)$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $4$ .

$A_{32} = - (12 - 4 \cdot 1)$

Multiply $- 4$ by $1$ .

$A_{32} = - (12 - 4)$

Simplify the expression.

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Subtract $4$ from $12$ .

$A_{32} = - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$A_{32} = - 8$

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

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

$A_{33} = (3) (4) - 4 \cdot 2$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $4$ .

$A_{33} = 12 - 4 \cdot 2$

Multiply $- 4$ by $2$ .

$A_{33} = 12 - 8$

Subtract $8$ from $12$ .

$A_{33} = 4$

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

$[\begin{array}{c} 4 & - 8 & 4 \\ - 4 & 8 & - 4 \\ 4 & - 8 & 4 \end{array}]$

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