Find the Inverse [[5,7],[6,9]]

$[\begin{array}{c} 5 & 7 \\ 6 & 9 \end{array}]$

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

$5 \cdot 9 - 6 \cdot 7$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $9$ .

$45 - 6 \cdot 7$

Multiply $- 6$ by $7$ .

$45 - 42$

Subtract $42$ from $45$ .

$3$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{3} [\begin{array}{c} 9 & - 7 \\ - 6 & 5 \end{array}]$

Step 4

Multiply $\frac{1}{3}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{3} \cdot 9 & \frac{1}{3} \cdot - 7 \\ \frac{1}{3} \cdot - 6 & \frac{1}{3} \cdot 5 \end{array}]$

Step 5

Simplify each element in the matrix.

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$[\begin{array}{c} \frac{1}{3} \cdot (3 (3)) & \frac{1}{3} \cdot - 7 \\ \frac{1}{3} \cdot - 6 & \frac{1}{3} \cdot 5 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{3} \cdot (3 \cdot 3) & \frac{1}{3} \cdot - 7 \\ \frac{1}{3} \cdot - 6 & \frac{1}{3} \cdot 5 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} 3 & \frac{1}{3} \cdot - 7 \\ \frac{1}{3} \cdot - 6 & \frac{1}{3} \cdot 5 \end{array}]$

Combine $\frac{1}{3}$ and $- 7$ .

$[\begin{array}{c} 3 & \frac{- 7}{3} \\ \frac{1}{3} \cdot - 6 & \frac{1}{3} \cdot 5 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} 3 & - \frac{7}{3} \\ \frac{1}{3} \cdot - 6 & \frac{1}{3} \cdot 5 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $- 6$ .

$[\begin{array}{c} 3 & - \frac{7}{3} \\ \frac{1}{3} \cdot (3 (- 2)) & \frac{1}{3} \cdot 5 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} 3 & - \frac{7}{3} \\ \frac{1}{3} \cdot (3 \cdot - 2) & \frac{1}{3} \cdot 5 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} 3 & - \frac{7}{3} \\ - 2 & \frac{1}{3} \cdot 5 \end{array}]$

Combine $\frac{1}{3}$ and $5$ .

$[\begin{array}{c} 3 & - \frac{7}{3} \\ - 2 & \frac{5}{3} \end{array}]$

Find the Inverse [[5,7],[6,9]]

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