Find the Inverse [[-5,-18],[2,7]]

$[\begin{array}{c} - 5 & - 18 \\ 2 & 7 \end{array}]$

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $| A |$ is the determinant of $A$ .

If $A = [\begin{array}{c} a & b \\ c & d \end{array}]$ then $A^{- 1} = \frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$

Find the determinant of $[\begin{array}{c} - 5 & - 18 \\ 2 & 7 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} - 5 & - 18 \\ 2 & 7 \end{array}] = | \begin{array}{c} - 5 & - 18 \\ 2 & 7 \end{array} |$

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

$(- 5) (7) - 2 \cdot - 18$

Simplify the determinant.

Simplify each term.

Multiply $- 5$ by $7$ .

$- 35 - 2 \cdot - 18$

Multiply $- 2$ by $- 18$ .

$- 35 + 36$

Add $- 35$ and $36$ .

$1$

Substitute the known values into the formula for the inverse of a matrix.

$\frac{1}{1} [\begin{array}{c} 7 & - (- 18) \\ - (2) & - 5 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 18)$ .

$\frac{1}{1} [\begin{array}{c} 7 & 18 \\ - (2) & - 5 \end{array}]$

Rearrange $- (2)$ .

$\frac{1}{1} [\begin{array}{c} 7 & 18 \\ - 2 & - 5 \end{array}]$

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

$[\begin{array}{c} \frac{1}{1} \cdot 7 & \frac{1}{1} \cdot 18 \\ \frac{1}{1} \cdot - 2 & \frac{1}{1} \cdot - 5 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{1} \cdot 7$ .

$[\begin{array}{c} 7 & \frac{1}{1} \cdot 18 \\ \frac{1}{1} \cdot - 2 & \frac{1}{1} \cdot - 5 \end{array}]$

Rearrange $\frac{1}{1} \cdot 18$ .

$[\begin{array}{c} 7 & 18 \\ \frac{1}{1} \cdot - 2 & \frac{1}{1} \cdot - 5 \end{array}]$

Rearrange $\frac{1}{1} \cdot - 2$ .

$[\begin{array}{c} 7 & 18 \\ - 2 & \frac{1}{1} \cdot - 5 \end{array}]$

Rearrange $\frac{1}{1} \cdot - 5$ .

$[\begin{array}{c} 7 & 18 \\ - 2 & - 5 \end{array}]$

Find the Inverse [[-5,-18],[2,7]]

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