Find the Inverse [[2,-9],[9,-40]]

$[\begin{array}{c} 2 & - 9 \\ 9 & - 40 \end{array}]$

Step 1

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 2

Find the determinant.

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

$2 \cdot - 40 - 9 \cdot - 9$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 40$ .

$- 80 - 9 \cdot - 9$

Multiply $- 9$ by $- 9$ .

$- 80 + 81$

Add $- 80$ and $81$ .

$1$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{1} [\begin{array}{c} - 40 & 9 \\ - 9 & 2 \end{array}]$

Step 4

Divide $1$ by $1$ .

$1 [\begin{array}{c} - 40 & 9 \\ - 9 & 2 \end{array}]$

Step 5

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 1 \cdot - 40 & 1 \cdot 9 \\ 1 \cdot - 9 & 1 \cdot 2 \end{array}]$

Step 6

Simplify each element in the matrix.

Multiply $- 40$ by $1$ .

$[\begin{array}{c} - 40 & 1 \cdot 9 \\ 1 \cdot - 9 & 1 \cdot 2 \end{array}]$

Multiply $9$ by $1$ .

$[\begin{array}{c} - 40 & 9 \\ 1 \cdot - 9 & 1 \cdot 2 \end{array}]$

Multiply $- 9$ by $1$ .

$[\begin{array}{c} - 40 & 9 \\ - 9 & 1 \cdot 2 \end{array}]$

Multiply $2$ by $1$ .

$[\begin{array}{c} - 40 & 9 \\ - 9 & 2 \end{array}]$

Find the Inverse [[2,-9],[9,-40]]

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