Algebra Examples

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}]$

Step 1

Set up the determinant by breaking it into smaller components.

$1 | \begin{array}{c} 3 & 0 \\ 0 & 5 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 0 & 5 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Step 2

Find the determinant of $1 | \begin{array}{c} 3 & 0 \\ 0 & 5 \end{array} |$ .

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

$(3) (5) + 0 \cdot 0 + 0 | \begin{array}{c} 0 & 0 \\ 0 & 5 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $5$ .

$15 + 0 \cdot 0 + 0 | \begin{array}{c} 0 & 0 \\ 0 & 5 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Multiply $0$ by $0$ .

$15 + 0 + 0 | \begin{array}{c} 0 & 0 \\ 0 & 5 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Add $15$ and $0$ .

$15 + 0 | \begin{array}{c} 0 & 0 \\ 0 & 5 \end{array} | + 0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Step 3

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

$15 + 0 + 0 | \begin{array}{c} 0 & 0 \\ 3 & 0 \end{array} |$

Step 4

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

$15 + 0 + 0$

Step 5

Add $15$ and $0$ .

$15 + 0$

Step 6

Add $15$ and $0$ .

$15$