Algebra Examples

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

Step 1

Set up the determinant by breaking it into smaller components.

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

Step 2

Find the determinant of $1 | \begin{array}{c} 1 & 4 \\ 3 & 0 \end{array} |$ .

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

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

Simplify the determinant.

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Multiply $(1) (0) - 3 \cdot 4$ by $1$ .

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

Simplify each term.

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Multiply $0$ by $1$ .

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

Multiply $- 3$ by $4$ .

$0 - 12 + 0 | \begin{array}{c} 3 & 2 \\ 3 & 0 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 1 & 4 \end{array} |$

Subtract $12$ from $0$ .

$- 12 + 0 | \begin{array}{c} 3 & 2 \\ 3 & 0 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 1 & 4 \end{array} |$

Step 3

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

$- 12 + 0 + 2 | \begin{array}{c} 3 & 2 \\ 1 & 4 \end{array} |$

Step 4

Find the determinant of $2 | \begin{array}{c} 3 & 2 \\ 1 & 4 \end{array} |$ .

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

$- 12 + 0 + 2 ((3) (4) - 1 \cdot 2)$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $4$ .

$- 12 + 0 + 2 (12 - 1 \cdot 2)$

Multiply $- 1$ by $2$ .

$- 12 + 0 + 2 (12 - 2)$

Simplify the expression.

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Subtract $2$ from $12$ .

$- 12 + 0 + 2 \cdot 10$

Multiply $2$ by $10$ .

$- 12 + 0 + 20$

Step 5

Add $- 12$ and $0$ .

$- 12 + 20$

Step 6

Add $- 12$ and $20$ .

$8$