Find the Inverse [[4,-6],[8,12]]

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

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

$4 \cdot 12 - 8 \cdot - 6$

Simplify the determinant.

Simplify each term.

Multiply $4$ by $12$ .

$48 - 8 \cdot - 6$

Multiply $- 8$ by $- 6$ .

$48 + 48$

Add $48$ and $48$ .

$96$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{96} [\begin{array}{c} 12 & 6 \\ - 8 & 4 \end{array}]$

Step 4

Multiply $\frac{1}{96}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{96} \cdot 12 & \frac{1}{96} \cdot 6 \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Step 5

Simplify each element in the matrix.

Cancel the common factor of $12$ .

Factor $12$ out of $96$ .

$[\begin{array}{c} \frac{1}{12 (8)} \cdot 12 & \frac{1}{96} \cdot 6 \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{12 \cdot 8} \cdot 12 & \frac{1}{96} \cdot 6 \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{96} \cdot 6 \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Cancel the common factor of $6$ .

Factor $6$ out of $96$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{6 (16)} \cdot 6 \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{6 \cdot 16} \cdot 6 \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ \frac{1}{96} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Cancel the common factor of $8$ .

Factor $8$ out of $96$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ \frac{1}{8 (12)} \cdot - 8 & \frac{1}{96} \cdot 4 \end{array}]$

Factor $8$ out of $- 8$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ \frac{1}{8 \cdot 12} \cdot (8 \cdot - 1) & \frac{1}{96} \cdot 4 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ \frac{1}{8 \cdot 12} \cdot (8 \cdot - 1) & \frac{1}{96} \cdot 4 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ \frac{1}{12} \cdot - 1 & \frac{1}{96} \cdot 4 \end{array}]$

Combine $\frac{1}{12}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ \frac{- 1}{12} & \frac{1}{96} \cdot 4 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ - \frac{1}{12} & \frac{1}{96} \cdot 4 \end{array}]$

Cancel the common factor of $4$ .

Factor $4$ out of $96$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ - \frac{1}{12} & \frac{1}{4 (24)} \cdot 4 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ - \frac{1}{12} & \frac{1}{4 \cdot 24} \cdot 4 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{16} \\ - \frac{1}{12} & \frac{1}{24} \end{array}]$

Find the Inverse [[4,-6],[8,12]]

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