Find the Determinant [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1

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

$1 | \begin{array}{c} 6 & 7 & 8 \\ 10 & 11 & 12 \\ 14 & 15 & 16 \end{array} | - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 6 & 7 & 8 \\ 10 & 11 & 12 \\ 14 & 15 & 16 \end{array} |$ .

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

$1 (6 | \begin{array}{c} 11 & 12 \\ 15 & 16 \end{array} | - 7 | \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} | + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 11 & 12 \\ 15 & 16 \end{array} |$ .

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

$1 (6 (11 \cdot 16 - 15 \cdot 12) - 7 | \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} | + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $11$ by $16$ .

$1 (6 (176 - 15 \cdot 12) - 7 | \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} | + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 15$ by $12$ .

$1 (6 (176 - 180) - 7 | \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} | + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $180$ from $176$ .

$1 (6 \cdot - 4 - 7 | \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} | + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} |$ .

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

$1 (6 \cdot - 4 - 7 (10 \cdot 16 - 14 \cdot 12) + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $10$ by $16$ .

$1 (6 \cdot - 4 - 7 (160 - 14 \cdot 12) + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 14$ by $12$ .

$1 (6 \cdot - 4 - 7 (160 - 168) + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $168$ from $160$ .

$1 (6 \cdot - 4 - 7 \cdot - 8 + 8 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |$ .

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

$1 (6 \cdot - 4 - 7 \cdot - 8 + 8 (10 \cdot 15 - 14 \cdot 11)) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $10$ by $15$ .

$1 (6 \cdot - 4 - 7 \cdot - 8 + 8 (150 - 14 \cdot 11)) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 14$ by $11$ .

$1 (6 \cdot - 4 - 7 \cdot - 8 + 8 (150 - 154)) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $154$ from $150$ .

$1 (6 \cdot - 4 - 7 \cdot - 8 + 8 \cdot - 4) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $- 4$ .

$1 (- 24 - 7 \cdot - 8 + 8 \cdot - 4) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 7$ by $- 8$ .

$1 (- 24 + 56 + 8 \cdot - 4) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $8$ by $- 4$ .

$1 (- 24 + 56 - 32) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Add $- 24$ and $56$ .

$1 (32 - 32) - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $32$ from $32$ .

$1 \cdot 0 - 2 | \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} | + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{array} |$ .

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

$1 \cdot 0 - 2 (5 | \begin{array}{c} 11 & 12 \\ 15 & 16 \end{array} | - 7 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 11 & 12 \\ 15 & 16 \end{array} |$ .

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

$1 \cdot 0 - 2 (5 (11 \cdot 16 - 15 \cdot 12) - 7 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $11$ by $16$ .

$1 \cdot 0 - 2 (5 (176 - 15 \cdot 12) - 7 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 15$ by $12$ .

$1 \cdot 0 - 2 (5 (176 - 180) - 7 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $180$ from $176$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} |$ .

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

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 (9 \cdot 16 - 13 \cdot 12) + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $16$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 (144 - 13 \cdot 12) + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 13$ by $12$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 (144 - 156) + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $156$ from $144$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 \cdot - 12 + 8 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |$ .

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

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 \cdot - 12 + 8 (9 \cdot 15 - 13 \cdot 11)) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $15$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 \cdot - 12 + 8 (135 - 13 \cdot 11)) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 13$ by $11$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 \cdot - 12 + 8 (135 - 143)) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $143$ from $135$ .

$1 \cdot 0 - 2 (5 \cdot - 4 - 7 \cdot - 12 + 8 \cdot - 8) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $- 4$ .

$1 \cdot 0 - 2 (- 20 - 7 \cdot - 12 + 8 \cdot - 8) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 7$ by $- 12$ .

$1 \cdot 0 - 2 (- 20 + 84 + 8 \cdot - 8) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $8$ by $- 8$ .

$1 \cdot 0 - 2 (- 20 + 84 - 64) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Add $- 20$ and $84$ .

$1 \cdot 0 - 2 (64 - 64) + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $64$ from $64$ .

$1 \cdot 0 - 2 \cdot 0 + 3 | \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} | - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 (5 | \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} | - 6 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 10 & 12 \\ 14 & 16 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 (5 (10 \cdot 16 - 14 \cdot 12) - 6 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $10$ by $16$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 (160 - 14 \cdot 12) - 6 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 14$ by $12$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 (160 - 168) - 6 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $168$ from $160$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 | \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} | + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 9 & 12 \\ 13 & 16 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 (9 \cdot 16 - 13 \cdot 12) + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $16$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 (144 - 13 \cdot 12) + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 13$ by $12$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 (144 - 156) + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $156$ from $144$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 \cdot - 12 + 8 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Evaluate $| \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 \cdot - 12 + 8 (9 \cdot 14 - 13 \cdot 10)) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $14$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 \cdot - 12 + 8 (126 - 13 \cdot 10)) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 13$ by $10$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 \cdot - 12 + 8 (126 - 130)) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $130$ from $126$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (5 \cdot - 8 - 6 \cdot - 12 + 8 \cdot - 4) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $- 8$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (- 40 - 6 \cdot - 12 + 8 \cdot - 4) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $- 6$ by $- 12$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (- 40 + 72 + 8 \cdot - 4) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Multiply $8$ by $- 4$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (- 40 + 72 - 32) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Add $- 40$ and $72$ .

$1 \cdot 0 - 2 \cdot 0 + 3 (32 - 32) - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Subtract $32$ from $32$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 | \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 | \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} | - 6 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} | + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Evaluate $| \begin{array}{c} 10 & 11 \\ 14 & 15 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 (10 \cdot 15 - 14 \cdot 11) - 6 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} | + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Simplify the determinant.

Simplify each term.

Multiply $10$ by $15$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 (150 - 14 \cdot 11) - 6 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} | + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Multiply $- 14$ by $11$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 (150 - 154) - 6 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} | + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Subtract $154$ from $150$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 | \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} | + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Evaluate $| \begin{array}{c} 9 & 11 \\ 13 & 15 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 (9 \cdot 15 - 13 \cdot 11) + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $15$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 (135 - 13 \cdot 11) + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Multiply $- 13$ by $11$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 (135 - 143) + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Subtract $143$ from $135$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 \cdot - 8 + 7 | \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |)$

Evaluate $| \begin{array}{c} 9 & 10 \\ 13 & 14 \end{array} |$ .

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

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 \cdot - 8 + 7 (9 \cdot 14 - 13 \cdot 10))$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $14$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 \cdot - 8 + 7 (126 - 13 \cdot 10))$

Multiply $- 13$ by $10$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 \cdot - 8 + 7 (126 - 130))$

Subtract $130$ from $126$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (5 \cdot - 4 - 6 \cdot - 8 + 7 \cdot - 4)$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $- 4$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (- 20 - 6 \cdot - 8 + 7 \cdot - 4)$

Multiply $- 6$ by $- 8$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (- 20 + 48 + 7 \cdot - 4)$

Multiply $7$ by $- 4$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (- 20 + 48 - 28)$

Add $- 20$ and $48$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 (28 - 28)$

Subtract $28$ from $28$ .

$1 \cdot 0 - 2 \cdot 0 + 3 \cdot 0 - 4 \cdot 0$

Step 6

Simplify the determinant.

Simplify each term.

Multiply $0$ by $1$ .

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

Multiply $- 2$ by $0$ .

$0 + 0 + 3 \cdot 0 - 4 \cdot 0$

Multiply $3$ by $0$ .

$0 + 0 + 0 - 4 \cdot 0$

Multiply $- 4$ by $0$ .

$0 + 0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Find the Determinant [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]

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