Find the Inverse [[3,-2],[2,6]]

$[\begin{array}{c} 3 & - 2 \\ 2 & 6 \end{array}]$

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

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

Simplify the determinant.

Simplify each term.

Multiply $3$ by $6$ .

$18 - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$18 + 4$

Add $18$ and $4$ .

$22$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{22} [\begin{array}{c} 6 & 2 \\ - 2 & 3 \end{array}]$

Step 4

Multiply $\frac{1}{22}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{22} \cdot 6 & \frac{1}{22} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Step 5

Simplify each element in the matrix.

Cancel the common factor of $2$ .

Factor $2$ out of $22$ .

$[\begin{array}{c} \frac{1}{2 (11)} \cdot 6 & \frac{1}{22} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Factor $2$ out of $6$ .

$[\begin{array}{c} \frac{1}{2 \cdot 11} \cdot (2 \cdot 3) & \frac{1}{22} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2 \cdot 11} \cdot (2 \cdot 3) & \frac{1}{22} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{11} \cdot 3 & \frac{1}{22} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Combine $\frac{1}{11}$ and $3$ .

$[\begin{array}{c} \frac{3}{11} & \frac{1}{22} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Cancel the common factor of $2$ .

Factor $2$ out of $22$ .

$[\begin{array}{c} \frac{3}{11} & \frac{1}{2 (11)} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{3}{11} & \frac{1}{2 \cdot 11} \cdot 2 \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ \frac{1}{22} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Cancel the common factor of $2$ .

Factor $2$ out of $22$ .

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ \frac{1}{2 (11)} \cdot - 2 & \frac{1}{22} \cdot 3 \end{array}]$

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ \frac{1}{2 \cdot 11} \cdot (2 \cdot - 1) & \frac{1}{22} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ \frac{1}{2 \cdot 11} \cdot (2 \cdot - 1) & \frac{1}{22} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ \frac{1}{11} \cdot - 1 & \frac{1}{22} \cdot 3 \end{array}]$

Combine $\frac{1}{11}$ and $- 1$ .

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ \frac{- 1}{11} & \frac{1}{22} \cdot 3 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ - \frac{1}{11} & \frac{1}{22} \cdot 3 \end{array}]$

Combine $\frac{1}{22}$ and $3$ .

$[\begin{array}{c} \frac{3}{11} & \frac{1}{11} \\ - \frac{1}{11} & \frac{3}{22} \end{array}]$

Find the Inverse [[3,-2],[2,6]]

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