Find the Determinant [[-4,5,6],[0,4,4],[-2,-5,4]]

$[\begin{array}{c} - 4 & 5 & 6 \\ 0 & 4 & 4 \\ - 2 & - 5 & 4 \end{array}]$

Step 1

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

$- 4 | \begin{array}{c} 4 & 4 \\ - 5 & 4 \end{array} | + 0 | \begin{array}{c} 5 & 6 \\ - 5 & 4 \end{array} | - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 5 & 6 \\ - 5 & 4 \end{array} |$ .

$- 4 | \begin{array}{c} 4 & 4 \\ - 5 & 4 \end{array} | + 0 - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Step 3

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

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

$- 4 (4 \cdot 4 - (- 5 \cdot 4)) + 0 - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $4$ by $4$ .

$- 4 (16 - (- 5 \cdot 4)) + 0 - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Multiply $- (- 5 \cdot 4)$ .

Multiply $- 5$ by $4$ .

$- 4 (16 - - 20) + 0 - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Multiply $- 1$ by $- 20$ .

$- 4 (16 + 20) + 0 - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Add $16$ and $20$ .

$- 4 \cdot 36 + 0 - 2 | \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 5 & 6 \\ 4 & 4 \end{array} |$ .

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

$- 4 \cdot 36 + 0 - 2 (5 \cdot 4 - 4 \cdot 6)$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $4$ .

$- 4 \cdot 36 + 0 - 2 (20 - 4 \cdot 6)$

Multiply $- 4$ by $6$ .

$- 4 \cdot 36 + 0 - 2 (20 - 24)$

Subtract $24$ from $20$ .

$- 4 \cdot 36 + 0 - 2 \cdot - 4$

Step 5

Simplify the determinant.

Simplify each term.

Multiply $- 4$ by $36$ .

$- 144 + 0 - 2 \cdot - 4$

Multiply $- 2$ by $- 4$ .

$- 144 + 0 + 8$

Add $- 144$ and $0$ .

$- 144 + 8$

Add $- 144$ and $8$ .

$- 136$

Find the Determinant [[-4,5,6],[0,4,4],[-2,-5,4]]

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