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

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

Step 1

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

$0 | \begin{array}{c} 2 & 4 \\ 0 & 1 \end{array} | + 2 | \begin{array}{c} 1 & 4 \\ - 2 & 1 \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ - 2 & 0 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 2 & 4 \\ 0 & 1 \end{array} |$ .

$0 + 2 | \begin{array}{c} 1 & 4 \\ - 2 & 1 \end{array} | - 3 | \begin{array}{c} 1 & 2 \\ - 2 & 0 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & 4 \\ - 2 & 1 \end{array} |$ .

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

$0 + 2 (1 \cdot 1 - (- 2 \cdot 4)) - 3 | \begin{array}{c} 1 & 2 \\ - 2 & 0 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $1$ by $1$ .

$0 + 2 (1 - (- 2 \cdot 4)) - 3 | \begin{array}{c} 1 & 2 \\ - 2 & 0 \end{array} |$

Multiply $- (- 2 \cdot 4)$ .

Multiply $- 2$ by $4$ .

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

Multiply $- 1$ by $- 8$ .

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

Add $1$ and $8$ .

$0 + 2 \cdot 9 - 3 | \begin{array}{c} 1 & 2 \\ - 2 & 0 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 1 & 2 \\ - 2 & 0 \end{array} |$ .

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

$0 + 2 \cdot 9 - 3 (1 \cdot 0 - (- 2 \cdot 2))$

Simplify the determinant.

Simplify each term.

Multiply $0$ by $1$ .

$0 + 2 \cdot 9 - 3 (0 - (- 2 \cdot 2))$

Multiply $- (- 2 \cdot 2)$ .

Multiply $- 2$ by $2$ .

$0 + 2 \cdot 9 - 3 (0 - - 4)$

Multiply $- 1$ by $- 4$ .

$0 + 2 \cdot 9 - 3 (0 + 4)$

Add $0$ and $4$ .

$0 + 2 \cdot 9 - 3 \cdot 4$

Step 5

Simplify the determinant.

Simplify each term.

Multiply $2$ by $9$ .

$0 + 18 - 3 \cdot 4$

Multiply $- 3$ by $4$ .

$0 + 18 - 12$

Add $0$ and $18$ .

$18 - 12$

Subtract $12$ from $18$ .

$6$

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

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