Finite Math Examples

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

Set up the determinant by breaking it into smaller components.

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

The determinant of $3 | \begin{array}{c} 4 & 2 \\ 2 & 1 \end{array} |$ is $0$ .

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

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

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $1$ .

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

Multiply $- 2$ by $2$ .

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

Simplify the expression.

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Subtract $4$ from $4$ .

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

Multiply $3$ by $0$ .

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

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

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

The determinant of $3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$ is $- 6$ .

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

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

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $2$ .

$0 + 0 + 3 (6 - 4 \cdot 2)$

Multiply $- 4$ by $2$ .

$0 + 0 + 3 (6 - 8)$

Simplify the expression.

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Subtract $8$ from $6$ .

$0 + 0 + 3 \cdot - 2$

Multiply $3$ by $- 2$ .

$0 + 0 - 6$

Add $0$ and $0$ .

$0 - 6$

Subtract $6$ from $0$ .

$- 6$

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