Find the Inverse [[3,5],[1,5]]

$[\begin{array}{c} 3 & 5 \\ 1 & 5 \end{array}]$

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 $[\begin{array}{c} 3 & 5 \\ 1 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} 3 & 5 \\ 1 & 5 \end{array}] = | \begin{array}{c} 3 & 5 \\ 1 & 5 \end{array} |$

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

$(3) (5) - 1 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $3$ by $5$ .

$15 - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$15 - 5$

Subtract $5$ from $15$ .

$10$

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

$\frac{1}{10} [\begin{array}{c} 5 & - (5) \\ - (1) & 3 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (5)$ .

$\frac{1}{10} [\begin{array}{c} 5 & - 5 \\ - (1) & 3 \end{array}]$

Rearrange $- (1)$ .

$\frac{1}{10} [\begin{array}{c} 5 & - 5 \\ - 1 & 3 \end{array}]$

Multiply $\frac{1}{10}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{10} \cdot 5 & \frac{1}{10} \cdot - 5 \\ \frac{1}{10} \cdot - 1 & \frac{1}{10} \cdot 3 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{10} \cdot 5$ .

$[\begin{array}{c} \frac{1}{2} & \frac{1}{10} \cdot - 5 \\ \frac{1}{10} \cdot - 1 & \frac{1}{10} \cdot 3 \end{array}]$

Rearrange $\frac{1}{10} \cdot - 5$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{2} \\ \frac{1}{10} \cdot - 1 & \frac{1}{10} \cdot 3 \end{array}]$

Rearrange $\frac{1}{10} \cdot - 1$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{10} & \frac{1}{10} \cdot 3 \end{array}]$

Rearrange $\frac{1}{10} \cdot 3$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{10} & \frac{3}{10} \end{array}]$

Find the Inverse [[3,5],[1,5]]

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