Find the Determinant [[seven , five , zero],[four , five , eight],[zero , minus one , five]]

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

Set up the determinant by breaking it into smaller components.

$7 | \begin{array}{c} 5 & 8 \\ - 1 & 5 \end{array} | - 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} | + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

The determinant of $7 | \begin{array}{c} 5 & 8 \\ - 1 & 5 \end{array} |$ is $231$ .

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

$7 ((5) (5) + 1 \cdot 8) - 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} | + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Clarify the determinant.

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Clarify each term.

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First, multiply $5$ by $5$ .

$7 (25 + 1 \cdot 8) - 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} | + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Multiply $8$ by $1$ .

$7 (25 + 8) - 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} | + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Clarify the expression.

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Add $25$ and $8$ .

$7 \cdot 33 - 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} | + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Multiply $7$ by $33$ .

$231 - 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} | + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

The determinant of $- 4 | \begin{array}{c} 5 & 0 \\ - 1 & 5 \end{array} |$ is $- 100$ .

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

$231 - 4 ((5) (5) + 1 \cdot 0) + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Clarify the determinant.

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Simplify each term.

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Multiply $5$ by $5$ .

$231 - 4 (25 + 1 \cdot 0) + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Initially, multiply $0$ by $1$ .

$231 - 4 (25 + 0) + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Simplify the expression.

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Add $25$ and $0$ .

$231 - 4 \cdot 25 + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

Multiply $- 4$ by $25$ .

$231 - 100 + 0 | \begin{array}{c} 5 & 0 \\ 5 & 8 \end{array} |$

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

$231 - 100 + 0$

Subtract $100$ from $231$ .

$131 + 0$

Add $131$ and $0$ .

$131$

Can't Add different size matrices.

$231 - 100 + 0$

The combined expressions are .

$131$

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