Find the Inverse [[-1/5,-2/5],[0,1/4]]

$[\begin{array}{c} - \frac{1}{5} & - \frac{2}{5} \\ 0 & \frac{1}{4} \end{array}]$

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

$- \frac{1}{5} \cdot \frac{1}{4} + 0 (- \frac{2}{5})$

Simplify the determinant.

Simplify each term.

Multiply $- \frac{1}{5} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{4}$ by $\frac{1}{5}$ .

$- \frac{1}{4 \cdot 5} + 0 (- \frac{2}{5})$

Multiply $4$ by $5$ .

$- \frac{1}{20} + 0 (- \frac{2}{5})$

Multiply $0 (- \frac{2}{5})$ .

Multiply $- 1$ by $0$ .

$- \frac{1}{20} + 0 (\frac{2}{5})$

Multiply $0$ by $\frac{2}{5}$ .

$- \frac{1}{20} + 0$

Add $- \frac{1}{20}$ and $0$ .

$- \frac{1}{20}$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{- \frac{1}{20}} [\begin{array}{c} \frac{1}{4} & \frac{2}{5} \\ 0 & - \frac{1}{5} \end{array}]$

Step 4

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{1}{20}} [\begin{array}{c} \frac{1}{4} & \frac{2}{5} \\ 0 & - \frac{1}{5} \end{array}]$

Move the negative in front of the fraction.

$- \frac{1}{\frac{1}{20}} [\begin{array}{c} \frac{1}{4} & \frac{2}{5} \\ 0 & - \frac{1}{5} \end{array}]$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 20) [\begin{array}{c} \frac{1}{4} & \frac{2}{5} \\ 0 & - \frac{1}{5} \end{array}]$

Step 6

Multiply $- (1 \cdot 20)$ .

Multiply $20$ by $1$ .

$- 1 \cdot 20 [\begin{array}{c} \frac{1}{4} & \frac{2}{5} \\ 0 & - \frac{1}{5} \end{array}]$

Multiply $- 1$ by $20$ .

$- 20 [\begin{array}{c} \frac{1}{4} & \frac{2}{5} \\ 0 & - \frac{1}{5} \end{array}]$

Step 7

Multiply $- 20$ by each element of the matrix.

$[\begin{array}{c} - 20 (\frac{1}{4}) & - 20 (\frac{2}{5}) \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Step 8

Simplify each element in the matrix.

Cancel the common factor of $4$ .

Factor $4$ out of $- 20$ .

$[\begin{array}{c} 4 (- 5) \frac{1}{4} & - 20 (\frac{2}{5}) \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Cancel the common factor.

$[\begin{array}{c} 4 \cdot - 5 \frac{1}{4} & - 20 (\frac{2}{5}) \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - 5 & - 20 (\frac{2}{5}) \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Cancel the common factor of $5$ .

Factor $5$ out of $- 20$ .

$[\begin{array}{c} - 5 & 5 (- 4) \frac{2}{5} \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - 5 & 5 \cdot - 4 \frac{2}{5} \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - 5 & - 4 \cdot 2 \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Multiply $- 4$ by $2$ .

$[\begin{array}{c} - 5 & - 8 \\ - 20 \cdot 0 & - 20 (- \frac{1}{5}) \end{array}]$

Multiply $- 20$ by $0$ .

$[\begin{array}{c} - 5 & - 8 \\ 0 & - 20 (- \frac{1}{5}) \end{array}]$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$[\begin{array}{c} - 5 & - 8 \\ 0 & - 20 (\frac{- 1}{5}) \end{array}]$

Factor $5$ out of $- 20$ .

$[\begin{array}{c} - 5 & - 8 \\ 0 & 5 (- 4) \frac{- 1}{5} \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - 5 & - 8 \\ 0 & 5 \cdot - 4 \frac{- 1}{5} \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - 5 & - 8 \\ 0 & - 4 \cdot - 1 \end{array}]$

Multiply $- 4$ by $- 1$ .

$[\begin{array}{c} - 5 & - 8 \\ 0 & 4 \end{array}]$

Find the Inverse [[-1/5,-2/5],[0,1/4]]

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