Find the Determinant [[0,7,7,3],[0,2,5,6],[5,7,9,4],[6,2,2,4]]

$[\begin{array}{c} 0 & 7 & 7 & 3 \\ 0 & 2 & 5 & 6 \\ 5 & 7 & 9 & 4 \\ 6 & 2 & 2 & 4 \end{array}]$

Step 1

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

$0 | \begin{array}{c} 2 & 5 & 6 \\ 7 & 9 & 4 \\ 2 & 2 & 4 \end{array} | + 0 | \begin{array}{c} 7 & 7 & 3 \\ 7 & 9 & 4 \\ 2 & 2 & 4 \end{array} | + 5 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 2 & 2 & 4 \end{array} | - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 2 & 5 & 6 \\ 7 & 9 & 4 \\ 2 & 2 & 4 \end{array} |$ .

$0 + 0 | \begin{array}{c} 7 & 7 & 3 \\ 7 & 9 & 4 \\ 2 & 2 & 4 \end{array} | + 5 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 2 & 2 & 4 \end{array} | - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Step 3

Multiply $0$ by $| \begin{array}{c} 7 & 7 & 3 \\ 7 & 9 & 4 \\ 2 & 2 & 4 \end{array} |$ .

$0 + 0 + 5 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 2 & 2 & 4 \end{array} | - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 2 & 2 & 4 \end{array} |$ .

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

$0 + 0 + 5 (7 | \begin{array}{c} 5 & 6 \\ 2 & 4 \end{array} | - 7 | \begin{array}{c} 2 & 6 \\ 2 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 2 & 2 \end{array} |) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Evaluate $| \begin{array}{c} 5 & 6 \\ 2 & 4 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $5$ by $4$ .

$0 + 0 + 5 (7 (20 - 2 \cdot 6) - 7 | \begin{array}{c} 2 & 6 \\ 2 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 2 & 2 \end{array} |) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Multiply $- 2$ by $6$ .

$0 + 0 + 5 (7 (20 - 12) - 7 | \begin{array}{c} 2 & 6 \\ 2 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 2 & 2 \end{array} |) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Subtract $12$ from $20$ .

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

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

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

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

Simplify the determinant.

Simplify each term.

Multiply $2$ by $4$ .

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

Multiply $- 2$ by $6$ .

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

Subtract $12$ from $8$ .

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

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

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

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

Simplify the determinant.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 2$ by $5$ .

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

Subtract $10$ from $4$ .

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

Simplify the determinant.

Simplify each term.

Multiply $7$ by $8$ .

$0 + 0 + 5 (56 - 7 \cdot - 4 + 3 \cdot - 6) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Multiply $- 7$ by $- 4$ .

$0 + 0 + 5 (56 + 28 + 3 \cdot - 6) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Multiply $3$ by $- 6$ .

$0 + 0 + 5 (56 + 28 - 18) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Add $56$ and $28$ .

$0 + 0 + 5 (84 - 18) - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Subtract $18$ from $84$ .

$0 + 0 + 5 \cdot 66 - 6 | \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 7 & 7 & 3 \\ 2 & 5 & 6 \\ 7 & 9 & 4 \end{array} |$ .

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

$0 + 0 + 5 \cdot 66 - 6 (7 | \begin{array}{c} 5 & 6 \\ 9 & 4 \end{array} | - 7 | \begin{array}{c} 2 & 6 \\ 7 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Evaluate $| \begin{array}{c} 5 & 6 \\ 9 & 4 \end{array} |$ .

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

$0 + 0 + 5 \cdot 66 - 6 (7 (5 \cdot 4 - 9 \cdot 6) - 7 | \begin{array}{c} 2 & 6 \\ 7 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $4$ .

$0 + 0 + 5 \cdot 66 - 6 (7 (20 - 9 \cdot 6) - 7 | \begin{array}{c} 2 & 6 \\ 7 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Multiply $- 9$ by $6$ .

$0 + 0 + 5 \cdot 66 - 6 (7 (20 - 54) - 7 | \begin{array}{c} 2 & 6 \\ 7 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Subtract $54$ from $20$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 | \begin{array}{c} 2 & 6 \\ 7 & 4 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Evaluate $| \begin{array}{c} 2 & 6 \\ 7 & 4 \end{array} |$ .

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

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 (2 \cdot 4 - 7 \cdot 6) + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $4$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 (8 - 7 \cdot 6) + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Multiply $- 7$ by $6$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 (8 - 42) + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Subtract $42$ from $8$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 \cdot - 34 + 3 | \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |)$

Evaluate $| \begin{array}{c} 2 & 5 \\ 7 & 9 \end{array} |$ .

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

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 \cdot - 34 + 3 (2 \cdot 9 - 7 \cdot 5))$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $9$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 \cdot - 34 + 3 (18 - 7 \cdot 5))$

Multiply $- 7$ by $5$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 \cdot - 34 + 3 (18 - 35))$

Subtract $35$ from $18$ .

$0 + 0 + 5 \cdot 66 - 6 (7 \cdot - 34 - 7 \cdot - 34 + 3 \cdot - 17)$

Simplify the determinant.

Simplify each term.

Multiply $7$ by $- 34$ .

$0 + 0 + 5 \cdot 66 - 6 (- 238 - 7 \cdot - 34 + 3 \cdot - 17)$

Multiply $- 7$ by $- 34$ .

$0 + 0 + 5 \cdot 66 - 6 (- 238 + 238 + 3 \cdot - 17)$

Multiply $3$ by $- 17$ .

$0 + 0 + 5 \cdot 66 - 6 (- 238 + 238 - 51)$

Add $- 238$ and $238$ .

$0 + 0 + 5 \cdot 66 - 6 (0 - 51)$

Subtract $51$ from $0$ .

$0 + 0 + 5 \cdot 66 - 6 \cdot - 51$

Step 6

Simplify the determinant.

Simplify each term.

Multiply $5$ by $66$ .

$0 + 0 + 330 - 6 \cdot - 51$

Multiply $- 6$ by $- 51$ .

$0 + 0 + 330 + 306$

Add $0$ and $0$ .

$0 + 330 + 306$

Add $0$ and $330$ .

$330 + 306$

Add $330$ and $306$ .

$636$

Find the Determinant [[0,7,7,3],[0,2,5,6],[5,7,9,4],[6,2,2,4]]

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