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

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

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

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

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

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

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

Simplify the determinant.

Simplify each term.

Multiply $5$ by $1$ .

$5 - 1 \cdot - 4$

Multiply $- 1$ by $- 4$ .

$5 + 4$

Add $5$ and $4$ .

$9$

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

$\frac{1}{9} [\begin{array}{c} 5 & - (- 4) \\ - (1) & 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 4)$ .

$\frac{1}{9} [\begin{array}{c} 5 & 4 \\ - (1) & 1 \end{array}]$

Rearrange $- (1)$ .

$\frac{1}{9} [\begin{array}{c} 5 & 4 \\ - 1 & 1 \end{array}]$

Multiply $\frac{1}{9}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{9} \cdot 5 & \frac{1}{9} \cdot 4 \\ \frac{1}{9} \cdot - 1 & \frac{1}{9} \cdot 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{9} \cdot 5$ .

$[\begin{array}{c} \frac{5}{9} & \frac{1}{9} \cdot 4 \\ \frac{1}{9} \cdot - 1 & \frac{1}{9} \cdot 1 \end{array}]$

Rearrange $\frac{1}{9} \cdot 4$ .

$[\begin{array}{c} \frac{5}{9} & \frac{4}{9} \\ \frac{1}{9} \cdot - 1 & \frac{1}{9} \cdot 1 \end{array}]$

Rearrange $\frac{1}{9} \cdot - 1$ .

$[\begin{array}{c} \frac{5}{9} & \frac{4}{9} \\ - \frac{1}{9} & \frac{1}{9} \cdot 1 \end{array}]$

Rearrange $\frac{1}{9} \cdot 1$ .

$[\begin{array}{c} \frac{5}{9} & \frac{4}{9} \\ - \frac{1}{9} & \frac{1}{9} \end{array}]$

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

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