Find the Inverse [[6,-9],[4,6]]

$[\begin{array}{c} 6 & - 9 \\ 4 & 6 \end{array}]$

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

$6 \cdot 6 - 4 \cdot - 9$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $6$ .

$36 - 4 \cdot - 9$

Multiply $- 4$ by $- 9$ .

$36 + 36$

Add $36$ and $36$ .

$72$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{72} [\begin{array}{c} 6 & 9 \\ - 4 & 6 \end{array}]$

Step 4

Multiply $\frac{1}{72}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{72} \cdot 6 & \frac{1}{72} \cdot 9 \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Step 5

Simplify each element in the matrix.

Cancel the common factor of $6$ .

Factor $6$ out of $72$ .

$[\begin{array}{c} \frac{1}{6 (12)} \cdot 6 & \frac{1}{72} \cdot 9 \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{6 \cdot 12} \cdot 6 & \frac{1}{72} \cdot 9 \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{72} \cdot 9 \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Cancel the common factor of $9$ .

Factor $9$ out of $72$ .

$[\begin{array}{c} \frac{1}{12} & \frac{1}{9 (8)} \cdot 9 \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{9 \cdot 8} \cdot 9 \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ \frac{1}{72} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Cancel the common factor of $4$ .

Factor $4$ out of $72$ .

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ \frac{1}{4 (18)} \cdot - 4 & \frac{1}{72} \cdot 6 \end{array}]$

Factor $4$ out of $- 4$ .

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ \frac{1}{4 \cdot 18} \cdot (4 \cdot - 1) & \frac{1}{72} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ \frac{1}{4 \cdot 18} \cdot (4 \cdot - 1) & \frac{1}{72} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ \frac{1}{18} \cdot - 1 & \frac{1}{72} \cdot 6 \end{array}]$

Combine $\frac{1}{18}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ \frac{- 1}{18} & \frac{1}{72} \cdot 6 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ - \frac{1}{18} & \frac{1}{72} \cdot 6 \end{array}]$

Cancel the common factor of $6$ .

Factor $6$ out of $72$ .

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ - \frac{1}{18} & \frac{1}{6 (12)} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ - \frac{1}{18} & \frac{1}{6 \cdot 12} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{12} & \frac{1}{8} \\ - \frac{1}{18} & \frac{1}{12} \end{array}]$

Find the Inverse [[6,-9],[4,6]]

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