Find the Inverse [[4,5],[3,6]]

$[\begin{array}{c} 4 & 5 \\ 3 & 6 \end{array}]$

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

$4 \cdot 6 - 3 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $4$ by $6$ .

$24 - 3 \cdot 5$

Multiply $- 3$ by $5$ .

$24 - 15$

Subtract $15$ from $24$ .

$9$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{9} [\begin{array}{c} 6 & - 5 \\ - 3 & 4 \end{array}]$

Step 4

Multiply $\frac{1}{9}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{9} \cdot 6 & \frac{1}{9} \cdot - 5 \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Step 5

Simplify each element in the matrix.

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$[\begin{array}{c} \frac{1}{3 (3)} \cdot 6 & \frac{1}{9} \cdot - 5 \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Factor $3$ out of $6$ .

$[\begin{array}{c} \frac{1}{3 \cdot 3} \cdot (3 \cdot 2) & \frac{1}{9} \cdot - 5 \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{3 \cdot 3} \cdot (3 \cdot 2) & \frac{1}{9} \cdot - 5 \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{3} \cdot 2 & \frac{1}{9} \cdot - 5 \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Combine $\frac{1}{3}$ and $2$ .

$[\begin{array}{c} \frac{2}{3} & \frac{1}{9} \cdot - 5 \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Combine $\frac{1}{9}$ and $- 5$ .

$[\begin{array}{c} \frac{2}{3} & \frac{- 5}{9} \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ \frac{1}{9} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ \frac{1}{3 (3)} \cdot - 3 & \frac{1}{9} \cdot 4 \end{array}]$

Factor $3$ out of $- 3$ .

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ \frac{1}{3 \cdot 3} \cdot (3 \cdot - 1) & \frac{1}{9} \cdot 4 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ \frac{1}{3 \cdot 3} \cdot (3 \cdot - 1) & \frac{1}{9} \cdot 4 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ \frac{1}{3} \cdot - 1 & \frac{1}{9} \cdot 4 \end{array}]$

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

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ \frac{- 1}{3} & \frac{1}{9} \cdot 4 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ - \frac{1}{3} & \frac{1}{9} \cdot 4 \end{array}]$

Combine $\frac{1}{9}$ and $4$ .

$[\begin{array}{c} \frac{2}{3} & - \frac{5}{9} \\ - \frac{1}{3} & \frac{4}{9} \end{array}]$

Find the Inverse [[4,5],[3,6]]

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