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

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

Step 1

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 2

Find the determinant.

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

$4 \cdot 2 - 2 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $4$ by $2$ .

$8 - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$8 - 6$

Subtract $6$ from $8$ .

$2$

Step 3

Substitute the known values into the formula for the inverse.

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

Step 4

Multiply $\frac{1}{2}$ by each element of the matrix.

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

Step 5

Simplify each element in the matrix.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{1}{2}$ and $- 3$ .

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

Move the negative in front of the fraction.

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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