Find the Inverse [[-3,-2],[5,3]]

$[\begin{array}{c} - 3 & - 2 \\ 5 & 3 \end{array}]$

Step 1

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

Step 2

Find the determinant of $[\begin{array}{c} - 3 & - 2 \\ 5 & 3 \end{array}]$ .

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

$determinant [\begin{array}{c} - 3 & - 2 \\ 5 & 3 \end{array}] = | \begin{array}{c} - 3 & - 2 \\ 5 & 3 \end{array} |$

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

$(- 3) (3) - 5 \cdot - 2$

Simplify the determinant.

Simplify each term.

Multiply $- 3$ by $3$ .

$- 9 - 5 \cdot - 2$

Multiply $- 5$ by $- 2$ .

$- 9 + 10$

Add $- 9$ and $10$ .

$1$

Step 3

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

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

Step 4

Simplify each element in the matrix.

Rearrange $- (- 2)$ .

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

Rearrange $- (5)$ .

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

Step 5

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

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

Step 6

Simplify each element in the matrix.

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

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

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

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

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

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

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

$[\begin{array}{c} 3 & 2 \\ - 5 & - 3 \end{array}]$

Find the Inverse [[-3,-2],[5,3]]

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