Find the Determinant [[-1,0,8],[7,3,4],[2,2,5]]

$[\begin{array}{c} - 1 & 0 & 8 \\ 7 & 3 & 4 \\ 2 & 2 & 5 \end{array}]$

Step 1

Multiply every element in the $1 s t$ row by its cofactor and add.

$- | \begin{array}{c} 3 & 4 \\ 2 & 5 \end{array} | + 0 | \begin{array}{c} 7 & 4 \\ 2 & 5 \end{array} | + 8 | \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 7 & 4 \\ 2 & 5 \end{array} |$ .

$- | \begin{array}{c} 3 & 4 \\ 2 & 5 \end{array} | + 0 + 8 | \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 3 & 4 \\ 2 & 5 \end{array} |$ .

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

$- (3 \cdot 5 - 2 \cdot 4) + 0 + 8 | \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $5$ .

$- (15 - 2 \cdot 4) + 0 + 8 | \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$

Multiply $- 2$ by $4$ .

$- (15 - 8) + 0 + 8 | \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$

Subtract $8$ from $15$ .

$- 1 \cdot 7 + 0 + 8 | \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 7 & 3 \\ 2 & 2 \end{array} |$ .

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

$- 1 \cdot 7 + 0 + 8 (7 \cdot 2 - 2 \cdot 3)$

Simplify the determinant.

Simplify each term.

Multiply $7$ by $2$ .

$- 1 \cdot 7 + 0 + 8 (14 - 2 \cdot 3)$

Multiply $- 2$ by $3$ .

$- 1 \cdot 7 + 0 + 8 (14 - 6)$

Subtract $6$ from $14$ .

$- 1 \cdot 7 + 0 + 8 \cdot 8$

Step 5

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $7$ .

$- 7 + 0 + 8 \cdot 8$

Multiply $8$ by $8$ .

$- 7 + 0 + 64$

Add $- 7$ and $0$ .

$- 7 + 64$

Add $- 7$ and $64$ .

$57$

Find the Determinant [[-1,0,8],[7,3,4],[2,2,5]]

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