Find the Inverse [[-5,9],[-7,12]]

$[\begin{array}{c} - 5 & 9 \\ - 7 & 12 \end{array}]$

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

$- 5 \cdot 12 - (- 7 \cdot 9)$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $12$ .

$- 60 - (- 7 \cdot 9)$

Multiply $- (- 7 \cdot 9)$ .

Multiply $- 7$ by $9$ .

$- 60 - - 63$

Multiply $- 1$ by $- 63$ .

$- 60 + 63$

Add $- 60$ and $63$ .

$3$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{3} [\begin{array}{c} 12 & - 9 \\ 7 & - 5 \end{array}]$

Step 4

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

$[\begin{array}{c} \frac{1}{3} \cdot 12 & \frac{1}{3} \cdot - 9 \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

Step 5

Simplify each element in the matrix.

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$[\begin{array}{c} \frac{1}{3} \cdot (3 (4)) & \frac{1}{3} \cdot - 9 \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{3} \cdot (3 \cdot 4) & \frac{1}{3} \cdot - 9 \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} 4 & \frac{1}{3} \cdot - 9 \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $- 9$ .

$[\begin{array}{c} 4 & \frac{1}{3} \cdot (3 (- 3)) \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} 4 & \frac{1}{3} \cdot (3 \cdot - 3) \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} 4 & - 3 \\ \frac{1}{3} \cdot 7 & \frac{1}{3} \cdot - 5 \end{array}]$

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

$[\begin{array}{c} 4 & - 3 \\ \frac{7}{3} & \frac{1}{3} \cdot - 5 \end{array}]$

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

$[\begin{array}{c} 4 & - 3 \\ \frac{7}{3} & \frac{- 5}{3} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} 4 & - 3 \\ \frac{7}{3} & - \frac{5}{3} \end{array}]$

Find the Inverse [[-5,9],[-7,12]]

Related Questions

Support LCMGCF.com by disabling your adblocker.