Find the Determinant A=[[1,2,-2],[2,1,-4],[3,0,-6]]

$A = [\begin{array}{c} 1 & 2 & - 2 \\ 2 & 1 & - 4 \\ 3 & 0 & - 6 \end{array}]$

Step 1

Set up the determinant by breaking it into smaller components.

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

Step 2

Find the determinant of $- 2 | \begin{array}{c} 2 & - 4 \\ 3 & - 6 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 6$ .

$- 2 (- 12 - 3 \cdot - 4) + 1 | \begin{array}{c} 1 & - 2 \\ 3 & - 6 \end{array} | + 0 | \begin{array}{c} 1 & - 2 \\ 2 & - 4 \end{array} |$

Multiply $- 3$ by $- 4$ .

$- 2 (- 12 + 12) + 1 | \begin{array}{c} 1 & - 2 \\ 3 & - 6 \end{array} | + 0 | \begin{array}{c} 1 & - 2 \\ 2 & - 4 \end{array} |$

Simplify the expression.

Add $- 12$ and $12$ .

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

Multiply $- 2$ by $0$ .

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

Step 3

Find the determinant of $1 | \begin{array}{c} 1 & - 2 \\ 3 & - 6 \end{array} |$ .

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

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

Simplify the determinant.

Simplify each term.

Multiply $- 6$ by $1$ .

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

Multiply $- 3$ by $- 2$ .

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

Add $- 6$ and $6$ .

$0 + 0 + 0 | \begin{array}{c} 1 & - 2 \\ 2 & - 4 \end{array} |$

Step 4

Since the matrix is multiplied by $0$ , the determinant is $0$ .

$0 + 0 + 0$

Step 5

Add $0$ and $0$ .

$0 + 0$

Step 6

Add $0$ and $0$ .

$0$

Find the Determinant A=[[1,2,-2],[2,1,-4],[3,0,-6]]

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