Find the Determinant [[1,3,2],[1,3,3],[2,7,8]]

$[\begin{array}{c} 1 & 3 & 2 \\ 1 & 3 & 3 \\ 2 & 7 & 8 \end{array}]$

Set up the determinant by breaking it into smaller components.

$1 | \begin{array}{c} 3 & 3 \\ 7 & 8 \end{array} | - 1 | \begin{array}{c} 3 & 2 \\ 7 & 8 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Find the determinant of $1 | \begin{array}{c} 3 & 3 \\ 7 & 8 \end{array} |$ .

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

$(3) (8) - 7 \cdot 3 - 1 | \begin{array}{c} 3 & 2 \\ 7 & 8 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $8$ .

$24 - 7 \cdot 3 - 1 | \begin{array}{c} 3 & 2 \\ 7 & 8 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Multiply $- 7$ by $3$ .

$24 - 21 - 1 | \begin{array}{c} 3 & 2 \\ 7 & 8 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Subtract $21$ from $24$ .

$3 - 1 | \begin{array}{c} 3 & 2 \\ 7 & 8 \end{array} | + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Find the determinant of $- 1 | \begin{array}{c} 3 & 2 \\ 7 & 8 \end{array} |$ .

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

$3 - 1 ((3) (8) - 7 \cdot 2) + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $8$ .

$3 - 1 (24 - 7 \cdot 2) + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Multiply $- 7$ by $2$ .

$3 - 1 (24 - 14) + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Simplify the expression.

Subtract $14$ from $24$ .

$3 - 1 \cdot 10 + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

Multiply $- 1$ by $10$ .

$3 - 10 + 2 | \begin{array}{c} 3 & 2 \\ 3 & 3 \end{array} |$

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

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

$3 - 10 + 2 ((3) (3) - 3 \cdot 2)$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $3$ .

$3 - 10 + 2 (9 - 3 \cdot 2)$

Multiply $- 3$ by $2$ .

$3 - 10 + 2 (9 - 6)$

Simplify the expression.

Subtract $6$ from $9$ .

$3 - 10 + 2 \cdot 3$

Multiply $2$ by $3$ .

$3 - 10 + 6$

Subtract $10$ from $3$ .

$- 7 + 6$

Add $- 7$ and $6$ .

$- 1$

Find the Determinant [[1,3,2],[1,3,3],[2,7,8]]

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