Linear Algebra Examples

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

Step 1

Refer to the corresponding sign matrix below.

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

Step 2

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

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Find the determinant of $[\begin{array}{c} 4 & 3 \\ - 4 & 8 \end{array}]$ .

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

$A_{11} = (4) (8) + 4 \cdot 3$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $8$ .

$A_{11} = 32 + 4 \cdot 3$

Multiply $4$ by $3$ .

$A_{11} = 32 + 12$

Add $32$ and $12$ .

$A_{11} = 44$

Find the determinant of $- [\begin{array}{c} 5 & 3 \\ 2 & 8 \end{array}]$ .

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

$A_{12} = - ((5) (8) - 2 \cdot 3)$

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $8$ .

$A_{12} = - (40 - 2 \cdot 3)$

Multiply $- 2$ by $3$ .

$A_{12} = - (40 - 6)$

Simplify the expression.

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Subtract $6$ from $40$ .

$A_{12} = - 1 \cdot 34$

Multiply $- 1$ by $34$ .

$A_{12} = - 34$

Find the determinant of $[\begin{array}{c} 5 & 4 \\ 2 & - 4 \end{array}]$ .

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

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

Simplify the determinant.

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Simplify each term.

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Multiply $5$ by $- 4$ .

$A_{13} = - 20 - 2 \cdot 4$

Multiply $- 2$ by $4$ .

$A_{13} = - 20 - 8$

Subtract $8$ from $- 20$ .

$A_{13} = - 28$

Find the determinant of $- [\begin{array}{c} 2 & - 1 \\ - 4 & 8 \end{array}]$ .

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

$A_{21} = - ((2) (8) + 4 \cdot - 1)$

Simplify the determinant.

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Simplify each term.

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Multiply $2$ by $8$ .

$A_{21} = - (16 + 4 \cdot - 1)$

Multiply $4$ by $- 1$ .

$A_{21} = - (16 - 4)$

Simplify the expression.

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Subtract $4$ from $16$ .

$A_{21} = - 1 \cdot 12$

Multiply $- 1$ by $12$ .

$A_{21} = - 12$

Find the determinant of $[\begin{array}{c} 1 & - 1 \\ 2 & 8 \end{array}]$ .

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

$A_{22} = (1) (8) - 2 \cdot - 1$

Simplify the determinant.

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Simplify each term.

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Multiply $8$ by $1$ .

$A_{22} = 8 - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$A_{22} = 8 + 2$

Add $8$ and $2$ .

$A_{22} = 10$

Find the determinant of $- [\begin{array}{c} 1 & 2 \\ 2 & - 4 \end{array}]$ .

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

$A_{23} = - ((1) (- 4) - 2 \cdot 2)$

Simplify the determinant.

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Simplify each term.

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Multiply $- 4$ by $1$ .

$A_{23} = - (- 4 - 2 \cdot 2)$

Multiply $- 2$ by $2$ .

$A_{23} = - (- 4 - 4)$

Simplify the expression.

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Subtract $4$ from $- 4$ .

$A_{23} = 8$

Multiply $- 1$ by $- 8$ .

$A_{23} = 8$

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

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

$A_{31} = (2) (3) - 4 \cdot - 1$

Simplify the determinant.

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Simplify each term.

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Multiply $2$ by $3$ .

$A_{31} = 6 - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$A_{31} = 6 + 4$

Add $6$ and $4$ .

$A_{31} = 10$

Find the determinant of $- [\begin{array}{c} 1 & - 1 \\ 5 & 3 \end{array}]$ .

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

$A_{32} = - ((1) (3) - 5 \cdot - 1)$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $1$ .

$A_{32} = - (3 - 5 \cdot - 1)$

Multiply $- 5$ by $- 1$ .

$A_{32} = - (3 + 5)$

Simplify the expression.

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Add $3$ and $5$ .

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

Multiply $- 1$ by $8$ .

$A_{32} = - 8$

Find the determinant of $[\begin{array}{c} 1 & 2 \\ 5 & 4 \end{array}]$ .

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

$A_{33} = (1) (4) - 5 \cdot 2$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $1$ .

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

Multiply $- 5$ by $2$ .

$A_{33} = 4 - 10$

Subtract $10$ from $4$ .

$A_{33} = - 6$

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

$[\begin{array}{c} 44 & - 34 & - 28 \\ - 12 & 10 & 8 \\ 10 & - 8 & - 6 \end{array}]$

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