Find the Determinant [[1,2,3],[4,5,6],[7,8,9]]

$[\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}]$

Step 1

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

$1 | \begin{array}{c} 5 & 6 \\ 8 & 9 \end{array} | - 2 | \begin{array}{c} 4 & 6 \\ 7 & 9 \end{array} | + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Step 2

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

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

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

Simplify the determinant.

Simplify each term.

Multiply $5$ by $9$ .

$1 (45 - 8 \cdot 6) - 2 | \begin{array}{c} 4 & 6 \\ 7 & 9 \end{array} | + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Multiply $- 8$ by $6$ .

$1 (45 - 48) - 2 | \begin{array}{c} 4 & 6 \\ 7 & 9 \end{array} | + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Subtract $48$ from $45$ .

$1 \cdot - 3 - 2 | \begin{array}{c} 4 & 6 \\ 7 & 9 \end{array} | + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Step 3

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

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

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

Simplify the determinant.

Simplify each term.

Multiply $4$ by $9$ .

$1 \cdot - 3 - 2 (36 - 7 \cdot 6) + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Multiply $- 7$ by $6$ .

$1 \cdot - 3 - 2 (36 - 42) + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Subtract $42$ from $36$ .

$1 \cdot - 3 - 2 \cdot - 6 + 3 | \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 4 & 5 \\ 7 & 8 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $4$ by $8$ .

$1 \cdot - 3 - 2 \cdot - 6 + 3 (32 - 7 \cdot 5)$

Multiply $- 7$ by $5$ .

$1 \cdot - 3 - 2 \cdot - 6 + 3 (32 - 35)$

Subtract $35$ from $32$ .

$1 \cdot - 3 - 2 \cdot - 6 + 3 \cdot - 3$

Step 5

Simplify the determinant.

Simplify each term.

Multiply $- 3$ by $1$ .

$- 3 - 2 \cdot - 6 + 3 \cdot - 3$

Multiply $- 2$ by $- 6$ .

$- 3 + 12 + 3 \cdot - 3$

Multiply $3$ by $- 3$ .

$- 3 + 12 - 9$

Add $- 3$ and $12$ .

$9 - 9$

Subtract $9$ from $9$ .

$0$

Find the Determinant [[1,2,3],[4,5,6],[7,8,9]]

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