Algebra Examples

$[\begin{array}{c} 9 & 0 & 8 \\ - 12 & 8 & 0 \\ 10 & - 8 & 8 \end{array}]$

Set up the determinant by breaking it into smaller components.

$0 | \begin{array}{c} - 12 & 0 \\ 10 & 8 \end{array} | + 8 | \begin{array}{c} 9 & 8 \\ 10 & 8 \end{array} | + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Since the matrix is multiplied by $0$ , the determinant is $0$ .

$0 + 8 | \begin{array}{c} 9 & 8 \\ 10 & 8 \end{array} | + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Find the determinant of $8 | \begin{array}{c} 9 & 8 \\ 10 & 8 \end{array} |$ .

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

$0 + 8 ((9) (8) - 10 \cdot 8) + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $9$ by $8$ .

$0 + 8 (72 - 10 \cdot 8) + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Multiply $- 10$ by $8$ .

$0 + 8 (72 - 80) + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Simplify the expression.

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Subtract $80$ from $72$ .

$0 + 8 \cdot - 8 + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Multiply $8$ by $- 8$ .

$0 - 64 + 8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$

Find the determinant of $8 | \begin{array}{c} 9 & 8 \\ - 12 & 0 \end{array} |$ .

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

$0 - 64 + 8 ((9) (0) + 12 \cdot 8)$

Simplify the determinant.

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Simplify each term.

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Multiply $9$ by $0$ .

$0 - 64 + 8 (0 + 12 \cdot 8)$

Multiply $12$ by $8$ .

$0 - 64 + 8 (0 + 96)$

Simplify the expression.

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Add $0$ and $96$ .

$0 - 64 + 8 \cdot 96$

Multiply $8$ by $96$ .

$0 - 64 + 768$

Subtract $64$ from $0$ .

$- 64 + 768$

Add $- 64$ and $768$ .

$704$