Find the Determinant [[20,9,8],[3,12,-6],[-19,4,11]]

$[\begin{array}{c} 20 & 9 & 8 \\ 3 & 12 & - 6 \\ - 19 & 4 & 11 \end{array}]$

Step 1

Multiply every element in the $1 s t$ row by its cofactor and add.

$20 | \begin{array}{c} 12 & - 6 \\ 4 & 11 \end{array} | - 9 | \begin{array}{c} 3 & - 6 \\ - 19 & 11 \end{array} | + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 12 & - 6 \\ 4 & 11 \end{array} |$ .

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

$20 (12 \cdot 11 - 4 \cdot - 6) - 9 | \begin{array}{c} 3 & - 6 \\ - 19 & 11 \end{array} | + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $12$ by $11$ .

$20 (132 - 4 \cdot - 6) - 9 | \begin{array}{c} 3 & - 6 \\ - 19 & 11 \end{array} | + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Multiply $- 4$ by $- 6$ .

$20 (132 + 24) - 9 | \begin{array}{c} 3 & - 6 \\ - 19 & 11 \end{array} | + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Add $132$ and $24$ .

$20 \cdot 156 - 9 | \begin{array}{c} 3 & - 6 \\ - 19 & 11 \end{array} | + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 3 & - 6 \\ - 19 & 11 \end{array} |$ .

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

$20 \cdot 156 - 9 (3 \cdot 11 - (- 19 \cdot - 6)) + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $11$ .

$20 \cdot 156 - 9 (33 - (- 19 \cdot - 6)) + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Multiply $- (- 19 \cdot - 6)$ .

Multiply $- 19$ by $- 6$ .

$20 \cdot 156 - 9 (33 - 1 \cdot 114) + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Multiply $- 1$ by $114$ .

$20 \cdot 156 - 9 (33 - 114) + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Subtract $114$ from $33$ .

$20 \cdot 156 - 9 \cdot - 81 + 8 | \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 3 & 12 \\ - 19 & 4 \end{array} |$ .

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

$20 \cdot 156 - 9 \cdot - 81 + 8 (3 \cdot 4 - (- 19 \cdot 12))$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $4$ .

$20 \cdot 156 - 9 \cdot - 81 + 8 (12 - (- 19 \cdot 12))$

Multiply $- (- 19 \cdot 12)$ .

Multiply $- 19$ by $12$ .

$20 \cdot 156 - 9 \cdot - 81 + 8 (12 - - 228)$

Multiply $- 1$ by $- 228$ .

$20 \cdot 156 - 9 \cdot - 81 + 8 (12 + 228)$

Add $12$ and $228$ .

$20 \cdot 156 - 9 \cdot - 81 + 8 \cdot 240$

Step 5

Simplify the determinant.

Simplify each term.

Multiply $20$ by $156$ .

$3120 - 9 \cdot - 81 + 8 \cdot 240$

Multiply $- 9$ by $- 81$ .

$3120 + 729 + 8 \cdot 240$

Multiply $8$ by $240$ .

$3120 + 729 + 1920$

Add $3120$ and $729$ .

$3849 + 1920$

Add $3849$ and $1920$ .

$5769$

Find the Determinant [[20,9,8],[3,12,-6],[-19,4,11]]

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