Find the Inverse [[-1,-4],[5,5]]

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

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 $[\begin{array}{c} - 1 & - 4 \\ 5 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} - 1 & - 4 \\ 5 & 5 \end{array}] = | \begin{array}{c} - 1 & - 4 \\ 5 & 5 \end{array} |$

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

$(- 1) (5) - 5 \cdot - 4$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $5$ .

$- 5 - 5 \cdot - 4$

Multiply $- 5$ by $- 4$ .

$- 5 + 20$

Add $- 5$ and $20$ .

$15$

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

$\frac{1}{15} [\begin{array}{c} 5 & - (- 4) \\ - (5) & - 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 4)$ .

$\frac{1}{15} [\begin{array}{c} 5 & 4 \\ - (5) & - 1 \end{array}]$

Rearrange $- (5)$ .

$\frac{1}{15} [\begin{array}{c} 5 & 4 \\ - 5 & - 1 \end{array}]$

Multiply $\frac{1}{15}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{15} \cdot 5 & \frac{1}{15} \cdot 4 \\ \frac{1}{15} \cdot - 5 & \frac{1}{15} \cdot - 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{15} \cdot 5$ .

$[\begin{array}{c} \frac{1}{3} & \frac{1}{15} \cdot 4 \\ \frac{1}{15} \cdot - 5 & \frac{1}{15} \cdot - 1 \end{array}]$

Rearrange $\frac{1}{15} \cdot 4$ .

$[\begin{array}{c} \frac{1}{3} & \frac{4}{15} \\ \frac{1}{15} \cdot - 5 & \frac{1}{15} \cdot - 1 \end{array}]$

Rearrange $\frac{1}{15} \cdot - 5$ .

$[\begin{array}{c} \frac{1}{3} & \frac{4}{15} \\ - \frac{1}{3} & \frac{1}{15} \cdot - 1 \end{array}]$

Rearrange $\frac{1}{15} \cdot - 1$ .

$[\begin{array}{c} \frac{1}{3} & \frac{4}{15} \\ - \frac{1}{3} & - \frac{1}{15} \end{array}]$

Find the Inverse [[-1,-4],[5,5]]

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