Find the Inverse C=[[-9,7],[-3,-1]]

$C = [\begin{array}{c} - 9 & 7 \\ - 3 & - 1 \end{array}]$

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

$- 9 \cdot - 1 - (- 3 \cdot 7)$

Simplify the determinant.

Simplify each term.

Multiply $- 9$ by $- 1$ .

$9 - (- 3 \cdot 7)$

Multiply $- (- 3 \cdot 7)$ .

Multiply $- 3$ by $7$ .

$9 - - 21$

Multiply $- 1$ by $- 21$ .

$9 + 21$

Add $9$ and $21$ .

$30$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{30} [\begin{array}{c} - 1 & - 7 \\ 3 & - 9 \end{array}]$

Step 4

Multiply $\frac{1}{30}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{30} \cdot - 1 & \frac{1}{30} \cdot - 7 \\ \frac{1}{30} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Step 5

Simplify each element in the matrix.

Combine $\frac{1}{30}$ and $- 1$ .

$[\begin{array}{c} \frac{- 1}{30} & \frac{1}{30} \cdot - 7 \\ \frac{1}{30} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{30} & \frac{1}{30} \cdot - 7 \\ \frac{1}{30} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Combine $\frac{1}{30}$ and $- 7$ .

$[\begin{array}{c} - \frac{1}{30} & \frac{- 7}{30} \\ \frac{1}{30} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{30} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $30$ .

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{3 (10)} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{3 \cdot 10} \cdot 3 & \frac{1}{30} \cdot - 9 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & \frac{1}{30} \cdot - 9 \end{array}]$

Cancel the common factor of $3$ .

Factor $3$ out of $30$ .

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & \frac{1}{3 (10)} \cdot - 9 \end{array}]$

Factor $3$ out of $- 9$ .

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & \frac{1}{3 \cdot 10} \cdot (3 \cdot - 3) \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & \frac{1}{3 \cdot 10} \cdot (3 \cdot - 3) \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & \frac{1}{10} \cdot - 3 \end{array}]$

Combine $\frac{1}{10}$ and $- 3$ .

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & \frac{- 3}{10} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{30} & - \frac{7}{30} \\ \frac{1}{10} & - \frac{3}{10} \end{array}]$

Find the Inverse C=[[-9,7],[-3,-1]]

Related Questions

Support LCMGCF.com by disabling your adblocker.