Find the Inverse [[-1,2],[-3,-1]]

$[\begin{array}{c} - 1 & 2 \\ - 3 & - 1 \end{array}]$

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

$- - 1 - (- 3 \cdot 2)$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$1 - (- 3 \cdot 2)$

Multiply $- (- 3 \cdot 2)$ .

Multiply $- 3$ by $2$ .

$1 - - 6$

Multiply $- 1$ by $- 6$ .

$1 + 6$

Add $1$ and $6$ .

$7$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{7} [\begin{array}{c} - 1 & - 2 \\ 3 & - 1 \end{array}]$

Step 4

Multiply $\frac{1}{7}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{7} \cdot - 1 & \frac{1}{7} \cdot - 2 \\ \frac{1}{7} \cdot 3 & \frac{1}{7} \cdot - 1 \end{array}]$

Step 5

Simplify each element in the matrix.

Combine $\frac{1}{7}$ and $- 1$ .

$[\begin{array}{c} \frac{- 1}{7} & \frac{1}{7} \cdot - 2 \\ \frac{1}{7} \cdot 3 & \frac{1}{7} \cdot - 1 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{7} & \frac{1}{7} \cdot - 2 \\ \frac{1}{7} \cdot 3 & \frac{1}{7} \cdot - 1 \end{array}]$

Combine $\frac{1}{7}$ and $- 2$ .

$[\begin{array}{c} - \frac{1}{7} & \frac{- 2}{7} \\ \frac{1}{7} \cdot 3 & \frac{1}{7} \cdot - 1 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{7} & - \frac{2}{7} \\ \frac{1}{7} \cdot 3 & \frac{1}{7} \cdot - 1 \end{array}]$

Combine $\frac{1}{7}$ and $3$ .

$[\begin{array}{c} - \frac{1}{7} & - \frac{2}{7} \\ \frac{3}{7} & \frac{1}{7} \cdot - 1 \end{array}]$

Combine $\frac{1}{7}$ and $- 1$ .

$[\begin{array}{c} - \frac{1}{7} & - \frac{2}{7} \\ \frac{3}{7} & \frac{- 1}{7} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{7} & - \frac{2}{7} \\ \frac{3}{7} & - \frac{1}{7} \end{array}]$

Find the Inverse [[-1,2],[-3,-1]]

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