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

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

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

$3 \cdot 5 - 3 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $5$ .

$15 - 3 \cdot 3$

Multiply $- 3$ by $3$ .

$15 - 9$

Subtract $9$ from $15$ .

$6$

Step 3

Substitute the known values into the formula for the inverse.

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

Step 4

Multiply $\frac{1}{6}$ by each element of the matrix.

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

Step 5

Simplify each element in the matrix.

Combine $\frac{1}{6}$ and $5$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$[\begin{array}{c} \frac{5}{6} & \frac{1}{3 (2)} \cdot - 3 \\ \frac{1}{6} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

Factor $3$ out of $- 3$ .

$[\begin{array}{c} \frac{5}{6} & \frac{1}{3 \cdot 2} \cdot (3 \cdot - 1) \\ \frac{1}{6} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{5}{6} & \frac{1}{3 \cdot 2} \cdot (3 \cdot - 1) \\ \frac{1}{6} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{5}{6} & \frac{1}{2} \cdot - 1 \\ \frac{1}{6} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

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

$[\begin{array}{c} \frac{5}{6} & \frac{- 1}{2} \\ \frac{1}{6} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ \frac{1}{6} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ \frac{1}{3 (2)} \cdot - 3 & \frac{1}{6} \cdot 3 \end{array}]$

Factor $3$ out of $- 3$ .

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ \frac{1}{3 \cdot 2} \cdot (3 \cdot - 1) & \frac{1}{6} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ \frac{1}{3 \cdot 2} \cdot (3 \cdot - 1) & \frac{1}{6} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ \frac{1}{2} \cdot - 1 & \frac{1}{6} \cdot 3 \end{array}]$

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

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ \frac{- 1}{2} & \frac{1}{6} \cdot 3 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{6} \cdot 3 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{3 (2)} \cdot 3 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{3 \cdot 2} \cdot 3 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{5}{6} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{2} \end{array}]$

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

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