Precalculus Examples

$[\begin{array}{c} 0 & - 5 \\ - 6 & - 6 \end{array}]$

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.

Tap for more steps...

Step 2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 \cdot - 6 - (- 6 \cdot - 5)$

Step 2.2

Simplify the determinant.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $0$ by $- 6$ .

$0 - (- 6 \cdot - 5)$

Step 2.2.1.2

Multiply $- (- 6 \cdot - 5)$ .

Tap for more steps...

Step 2.2.1.2.1

Multiply $- 6$ by $- 5$ .

$0 - 1 \cdot 30$

Step 2.2.1.2.2

Multiply $- 1$ by $30$ .

$0 - 30$

Step 2.2.2

Subtract $30$ from $0$ .

$- 30$

Step 3

Since the determinant is non-zero, the inverse exists.

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 30} [\begin{array}{c} - 6 & 5 \\ 6 & 0 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{30} [\begin{array}{c} - 6 & 5 \\ 6 & 0 \end{array}]$

Step 6

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

$[\begin{array}{c} - \frac{1}{30} \cdot - 6 & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7

Simplify each element in the matrix.

Tap for more steps...

Step 7.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 7.1.1

Move the leading negative in $- \frac{1}{30}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{30} \cdot - 6 & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.1.2

Factor $6$ out of $30$ .

$[\begin{array}{c} \frac{- 1}{6 (5)} \cdot - 6 & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.1.3

Factor $6$ out of $- 6$ .

$[\begin{array}{c} \frac{- 1}{6 \cdot 5} \cdot (6 \cdot - 1) & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.1.4

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{6 \cdot 5} \cdot (6 \cdot - 1) & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.1.5

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{5} \cdot - 1 & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.2

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

$[\begin{array}{c} \frac{- - 1}{5} & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.3

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 7.4.1

Move the leading negative in $- \frac{1}{30}$ into the numerator.

$[\begin{array}{c} \frac{1}{5} & \frac{- 1}{30} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.4.2

Factor $5$ out of $30$ .

$[\begin{array}{c} \frac{1}{5} & \frac{- 1}{5 (6)} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.4.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{5} & \frac{- 1}{5 \cdot 6} \cdot 5 \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.4.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{5} & \frac{- 1}{6} \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.5

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ - \frac{1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.6

Cancel the common factor of $6$ .

Tap for more steps...

Step 7.6.1

Move the leading negative in $- \frac{1}{30}$ into the numerator.

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ \frac{- 1}{30} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.6.2

Factor $6$ out of $30$ .

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ \frac{- 1}{6 (5)} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.6.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ \frac{- 1}{6 \cdot 5} \cdot 6 & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.6.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ \frac{- 1}{5} & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ - \frac{1}{5} & - \frac{1}{30} \cdot 0 \end{array}]$

Step 7.8

Multiply $- \frac{1}{30} \cdot 0$ .

Tap for more steps...

Step 7.8.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ - \frac{1}{5} & 0 (\frac{1}{30}) \end{array}]$

Step 7.8.2

Multiply $0$ by $\frac{1}{30}$ .

$[\begin{array}{c} \frac{1}{5} & - \frac{1}{6} \\ - \frac{1}{5} & 0 \end{array}]$