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

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

Set up the determinant by breaking it into smaller components.

$- 1 | \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

The determinant of $- 1 | \begin{array}{c} - 1 & - 2 \\ 5 & 1 \end{array} |$ is $- 9$ .

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

$- 1 ((- 1) (1) - 5 \cdot - 2) + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $- 1$ by $1$ .

$- 1 (- 1 - 5 \cdot - 2) + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Multiply $- 5$ by $- 2$ .

$- 1 (- 1 + 10) + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Simplify the expression.

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Add $- 1$ and $10$ .

$- 1 \cdot 9 + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

Multiply $- 1$ by $9$ .

$- 9 + 0 | \begin{array}{c} - 4 & 2 \\ 5 & 1 \end{array} | + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

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

$- 9 + 0 + 0 | \begin{array}{c} - 4 & 2 \\ - 1 & - 2 \end{array} |$

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

$- 9 + 0 + 0$

Add $- 9$ and $0$ .

$- 9 + 0$

Add $- 9$ and $0$ .

$- 9$

Can’t combine different size matrices.

$- 9 + 0 + 0$

The combined expressions are .

$- 9$

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

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