Find the Inverse [[2,-3],[3,-4]]

$[\begin{array}{c} 2 & - 3 \\ 3 & - 4 \end{array}]$

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 $[\begin{array}{c} 2 & - 3 \\ 3 & - 4 \end{array}]$ .

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

$determinant [\begin{array}{c} 2 & - 3 \\ 3 & - 4 \end{array}] = | \begin{array}{c} 2 & - 3 \\ 3 & - 4 \end{array} |$

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

$(2) (- 4) - 3 \cdot - 3$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 4$ .

$- 8 - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$- 8 + 9$

Add $- 8$ and $9$ .

$1$

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

$\frac{1}{1} [\begin{array}{c} - 4 & - (- 3) \\ - (3) & 2 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 3)$ .

$\frac{1}{1} [\begin{array}{c} - 4 & 3 \\ - (3) & 2 \end{array}]$

Rearrange $- (3)$ .

$\frac{1}{1} [\begin{array}{c} - 4 & 3 \\ - 3 & 2 \end{array}]$

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

$[\begin{array}{c} \frac{1}{1} \cdot - 4 & \frac{1}{1} \cdot 3 \\ \frac{1}{1} \cdot - 3 & \frac{1}{1} \cdot 2 \end{array}]$

Simplify each element in the matrix.

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

$[\begin{array}{c} - 4 & \frac{1}{1} \cdot 3 \\ \frac{1}{1} \cdot - 3 & \frac{1}{1} \cdot 2 \end{array}]$

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

$[\begin{array}{c} - 4 & 3 \\ \frac{1}{1} \cdot - 3 & \frac{1}{1} \cdot 2 \end{array}]$

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

$[\begin{array}{c} - 4 & 3 \\ - 3 & \frac{1}{1} \cdot 2 \end{array}]$

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

$[\begin{array}{c} - 4 & 3 \\ - 3 & 2 \end{array}]$

Find the Inverse [[2,-3],[3,-4]]

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