Find the Inverse [[-2,5],[1,6]]

$[\begin{array}{c} - 2 & 5 \\ 1 & 6 \end{array}]$

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.

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

$- 2 \cdot 6 - 1 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $6$ .

$- 12 - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$- 12 - 5$

Subtract $5$ from $- 12$ .

$- 17$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{- 17} [\begin{array}{c} 6 & - 5 \\ - 1 & - 2 \end{array}]$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{17} [\begin{array}{c} 6 & - 5 \\ - 1 & - 2 \end{array}]$

Step 5

Multiply $- \frac{1}{17}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{17} \cdot 6 & - \frac{1}{17} \cdot - 5 \\ - \frac{1}{17} \cdot - 1 & - \frac{1}{17} \cdot - 2 \end{array}]$

Step 6

Simplify each element in the matrix.

Multiply $- \frac{1}{17} \cdot 6$ .

Multiply $6$ by $- 1$ .

$[\begin{array}{c} - 6 (\frac{1}{17}) & - \frac{1}{17} \cdot - 5 \\ - \frac{1}{17} \cdot - 1 & - \frac{1}{17} \cdot - 2 \end{array}]$

Combine $- 6$ and $\frac{1}{17}$ .

$[\begin{array}{c} \frac{- 6}{17} & - \frac{1}{17} \cdot - 5 \\ - \frac{1}{17} \cdot - 1 & - \frac{1}{17} \cdot - 2 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{6}{17} & - \frac{1}{17} \cdot - 5 \\ - \frac{1}{17} \cdot - 1 & - \frac{1}{17} \cdot - 2 \end{array}]$

Multiply $- \frac{1}{17} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} - \frac{6}{17} & 5 (\frac{1}{17}) \\ - \frac{1}{17} \cdot - 1 & - \frac{1}{17} \cdot - 2 \end{array}]$

Combine $5$ and $\frac{1}{17}$ .

$[\begin{array}{c} - \frac{6}{17} & \frac{5}{17} \\ - \frac{1}{17} \cdot - 1 & - \frac{1}{17} \cdot - 2 \end{array}]$

Multiply $- \frac{1}{17} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{6}{17} & \frac{5}{17} \\ 1 (\frac{1}{17}) & - \frac{1}{17} \cdot - 2 \end{array}]$

Multiply $\frac{1}{17}$ by $1$ .

$[\begin{array}{c} - \frac{6}{17} & \frac{5}{17} \\ \frac{1}{17} & - \frac{1}{17} \cdot - 2 \end{array}]$

Multiply $- \frac{1}{17} \cdot - 2$ .

Multiply $- 2$ by $- 1$ .

$[\begin{array}{c} - \frac{6}{17} & \frac{5}{17} \\ \frac{1}{17} & 2 (\frac{1}{17}) \end{array}]$

Combine $2$ and $\frac{1}{17}$ .

$[\begin{array}{c} - \frac{6}{17} & \frac{5}{17} \\ \frac{1}{17} & \frac{2}{17} \end{array}]$

Find the Inverse [[-2,5],[1,6]]

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