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

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

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

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

Simplify the determinant.

Simplify each term.

Multiply $3$ by $- 2$ .

$- 6 - 3 \cdot 2$

Multiply $- 3$ by $2$ .

$- 6 - 6$

Subtract $6$ from $- 6$ .

$- 12$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{- 12} [\begin{array}{c} - 2 & - 2 \\ - 3 & 3 \end{array}]$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{12} [\begin{array}{c} - 2 & - 2 \\ - 3 & 3 \end{array}]$

Step 5

Multiply $- \frac{1}{12}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{12} \cdot - 2 & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Step 6

Simplify each element in the matrix.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{12} \cdot - 2 & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Factor $2$ out of $12$ .

$[\begin{array}{c} \frac{- 1}{2 (6)} \cdot - 2 & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{- 1}{2 \cdot 6} \cdot (2 \cdot - 1) & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{2 \cdot 6} \cdot (2 \cdot - 1) & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{6} \cdot - 1 & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Combine $\frac{- 1}{6}$ and $- 1$ .

$[\begin{array}{c} \frac{- - 1}{6} & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{6} & - \frac{1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$[\begin{array}{c} \frac{1}{6} & \frac{- 1}{12} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Factor $2$ out of $12$ .

$[\begin{array}{c} \frac{1}{6} & \frac{- 1}{2 (6)} \cdot - 2 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{1}{6} & \frac{- 1}{2 \cdot 6} \cdot (2 \cdot - 1) \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{6} & \frac{- 1}{2 \cdot 6} \cdot (2 \cdot - 1) \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{6} & \frac{- 1}{6} \cdot - 1 \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Combine $\frac{- 1}{6}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{6} & \frac{- - 1}{6} \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ - \frac{1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{- 1}{12} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Factor $3$ out of $12$ .

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{- 1}{3 (4)} \cdot - 3 & - \frac{1}{12} \cdot 3 \end{array}]$

Factor $3$ out of $- 3$ .

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{- 1}{3 \cdot 4} \cdot (3 \cdot - 1) & - \frac{1}{12} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{- 1}{3 \cdot 4} \cdot (3 \cdot - 1) & - \frac{1}{12} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{- 1}{4} \cdot - 1 & - \frac{1}{12} \cdot 3 \end{array}]$

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

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{- - 1}{4} & - \frac{1}{12} \cdot 3 \end{array}]$

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{1}{4} & - \frac{1}{12} \cdot 3 \end{array}]$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{1}{4} & \frac{- 1}{12} \cdot 3 \end{array}]$

Factor $3$ out of $12$ .

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{1}{4} & \frac{- 1}{3 (4)} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{1}{4} & \frac{- 1}{3 \cdot 4} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{1}{4} & \frac{- 1}{4} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{6} & \frac{1}{6} \\ \frac{1}{4} & - \frac{1}{4} \end{array}]$

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

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