Find the Determinant [[0,6,1,9],[0,6,8,2],[3,9,5,6],[2,4,3,4]]

$[\begin{array}{c} 0 & 6 & 1 & 9 \\ 0 & 6 & 8 & 2 \\ 3 & 9 & 5 & 6 \\ 2 & 4 & 3 & 4 \end{array}]$

Step 1

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

$0 | \begin{array}{c} 6 & 8 & 2 \\ 9 & 5 & 6 \\ 4 & 3 & 4 \end{array} | + 0 | \begin{array}{c} 6 & 1 & 9 \\ 9 & 5 & 6 \\ 4 & 3 & 4 \end{array} | + 3 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 4 & 3 & 4 \end{array} | - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 6 & 8 & 2 \\ 9 & 5 & 6 \\ 4 & 3 & 4 \end{array} |$ .

$0 + 0 | \begin{array}{c} 6 & 1 & 9 \\ 9 & 5 & 6 \\ 4 & 3 & 4 \end{array} | + 3 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 4 & 3 & 4 \end{array} | - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Step 3

Multiply $0$ by $| \begin{array}{c} 6 & 1 & 9 \\ 9 & 5 & 6 \\ 4 & 3 & 4 \end{array} |$ .

$0 + 0 + 3 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 4 & 3 & 4 \end{array} | - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 4 & 3 & 4 \end{array} |$ .

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

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

Evaluate $| \begin{array}{c} 8 & 2 \\ 3 & 4 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $8$ by $4$ .

$0 + 0 + 3 (6 (32 - 3 \cdot 2) - | \begin{array}{c} 6 & 2 \\ 4 & 4 \end{array} | + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Multiply $- 3$ by $2$ .

$0 + 0 + 3 (6 (32 - 6) - | \begin{array}{c} 6 & 2 \\ 4 & 4 \end{array} | + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Subtract $6$ from $32$ .

$0 + 0 + 3 (6 \cdot 26 - | \begin{array}{c} 6 & 2 \\ 4 & 4 \end{array} | + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Evaluate $| \begin{array}{c} 6 & 2 \\ 4 & 4 \end{array} |$ .

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

$0 + 0 + 3 (6 \cdot 26 - (6 \cdot 4 - 4 \cdot 2) + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $4$ .

$0 + 0 + 3 (6 \cdot 26 - (24 - 4 \cdot 2) + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Multiply $- 4$ by $2$ .

$0 + 0 + 3 (6 \cdot 26 - (24 - 8) + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Subtract $8$ from $24$ .

$0 + 0 + 3 (6 \cdot 26 - 1 \cdot 16 + 9 | \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Evaluate $| \begin{array}{c} 6 & 8 \\ 4 & 3 \end{array} |$ .

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

$0 + 0 + 3 (6 \cdot 26 - 1 \cdot 16 + 9 (6 \cdot 3 - 4 \cdot 8)) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $3$ .

$0 + 0 + 3 (6 \cdot 26 - 1 \cdot 16 + 9 (18 - 4 \cdot 8)) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Multiply $- 4$ by $8$ .

$0 + 0 + 3 (6 \cdot 26 - 1 \cdot 16 + 9 (18 - 32)) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Subtract $32$ from $18$ .

$0 + 0 + 3 (6 \cdot 26 - 1 \cdot 16 + 9 \cdot - 14) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $26$ .

$0 + 0 + 3 (156 - 1 \cdot 16 + 9 \cdot - 14) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Multiply $- 1$ by $16$ .

$0 + 0 + 3 (156 - 16 + 9 \cdot - 14) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Multiply $9$ by $- 14$ .

$0 + 0 + 3 (156 - 16 - 126) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Subtract $16$ from $156$ .

$0 + 0 + 3 (140 - 126) - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Subtract $126$ from $140$ .

$0 + 0 + 3 \cdot 14 - 2 | \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 6 & 1 & 9 \\ 6 & 8 & 2 \\ 9 & 5 & 6 \end{array} |$ .

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

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

Evaluate $| \begin{array}{c} 8 & 2 \\ 5 & 6 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $8$ by $6$ .

$0 + 0 + 3 \cdot 14 - 2 (6 (48 - 5 \cdot 2) - | \begin{array}{c} 6 & 2 \\ 9 & 6 \end{array} | + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Multiply $- 5$ by $2$ .

$0 + 0 + 3 \cdot 14 - 2 (6 (48 - 10) - | \begin{array}{c} 6 & 2 \\ 9 & 6 \end{array} | + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Subtract $10$ from $48$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - | \begin{array}{c} 6 & 2 \\ 9 & 6 \end{array} | + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Evaluate $| \begin{array}{c} 6 & 2 \\ 9 & 6 \end{array} |$ .

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

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - (6 \cdot 6 - 9 \cdot 2) + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $6$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - (36 - 9 \cdot 2) + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Multiply $- 9$ by $2$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - (36 - 18) + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Subtract $18$ from $36$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - 1 \cdot 18 + 9 | \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |)$

Evaluate $| \begin{array}{c} 6 & 8 \\ 9 & 5 \end{array} |$ .

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

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - 1 \cdot 18 + 9 (6 \cdot 5 - 9 \cdot 8))$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $5$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - 1 \cdot 18 + 9 (30 - 9 \cdot 8))$

Multiply $- 9$ by $8$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - 1 \cdot 18 + 9 (30 - 72))$

Subtract $72$ from $30$ .

$0 + 0 + 3 \cdot 14 - 2 (6 \cdot 38 - 1 \cdot 18 + 9 \cdot - 42)$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $38$ .

$0 + 0 + 3 \cdot 14 - 2 (228 - 1 \cdot 18 + 9 \cdot - 42)$

Multiply $- 1$ by $18$ .

$0 + 0 + 3 \cdot 14 - 2 (228 - 18 + 9 \cdot - 42)$

Multiply $9$ by $- 42$ .

$0 + 0 + 3 \cdot 14 - 2 (228 - 18 - 378)$

Subtract $18$ from $228$ .

$0 + 0 + 3 \cdot 14 - 2 (210 - 378)$

Subtract $378$ from $210$ .

$0 + 0 + 3 \cdot 14 - 2 \cdot - 168$

Step 6

Simplify the determinant.

Simplify each term.

Multiply $3$ by $14$ .

$0 + 0 + 42 - 2 \cdot - 168$

Multiply $- 2$ by $- 168$ .

$0 + 0 + 42 + 336$

Add $0$ and $0$ .

$0 + 42 + 336$

Add $0$ and $42$ .

$42 + 336$

Add $42$ and $336$ .

$378$

Find the Determinant [[0,6,1,9],[0,6,8,2],[3,9,5,6],[2,4,3,4]]

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