Find the Inverse [[8,3],[9,4]]

$[\begin{array}{c} 8 & 3 \\ 9 & 4 \end{array}]$

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 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$8 \cdot 4 - 9 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $8$ by $4$ .

$32 - 9 \cdot 3$

Multiply $- 9$ by $3$ .

$32 - 27$

Subtract $27$ from $32$ .

$5$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{5} [\begin{array}{c} 4 & - 3 \\ - 9 & 8 \end{array}]$

Step 4

Multiply $\frac{1}{5}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{5} \cdot 4 & \frac{1}{5} \cdot - 3 \\ \frac{1}{5} \cdot - 9 & \frac{1}{5} \cdot 8 \end{array}]$

Step 5

Simplify each element in the matrix.

Combine $\frac{1}{5}$ and $4$ .

$[\begin{array}{c} \frac{4}{5} & \frac{1}{5} \cdot - 3 \\ \frac{1}{5} \cdot - 9 & \frac{1}{5} \cdot 8 \end{array}]$

Combine $\frac{1}{5}$ and $- 3$ .

$[\begin{array}{c} \frac{4}{5} & \frac{- 3}{5} \\ \frac{1}{5} \cdot - 9 & \frac{1}{5} \cdot 8 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{4}{5} & - \frac{3}{5} \\ \frac{1}{5} \cdot - 9 & \frac{1}{5} \cdot 8 \end{array}]$

Combine $\frac{1}{5}$ and $- 9$ .

$[\begin{array}{c} \frac{4}{5} & - \frac{3}{5} \\ \frac{- 9}{5} & \frac{1}{5} \cdot 8 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{4}{5} & - \frac{3}{5} \\ - \frac{9}{5} & \frac{1}{5} \cdot 8 \end{array}]$

Combine $\frac{1}{5}$ and $8$ .

$[\begin{array}{c} \frac{4}{5} & - \frac{3}{5} \\ - \frac{9}{5} & \frac{8}{5} \end{array}]$

Find the Inverse [[8,3],[9,4]]

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