Find the Inverse [[-4,2],[3,-1]]

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

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

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

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

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

$(- 4) (- 1) - 3 \cdot 2$

Simplify the determinant.

Simplify each term.

Multiply $- 4$ by $- 1$ .

$4 - 3 \cdot 2$

Multiply $- 3$ by $2$ .

$4 - 6$

Subtract $6$ from $4$ .

$- 2$

Step 3

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

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

Step 4

Simplify each element in the matrix.

Rearrange $- (2)$ .

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

Rearrange $- (3)$ .

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

Step 5

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

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

Step 6

Simplify each element in the matrix.

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

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

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

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

Rearrange $\frac{1}{- 2} \cdot - 3$ .

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

Rearrange $\frac{1}{- 2} \cdot - 4$ .

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

Find the Inverse [[-4,2],[3,-1]]

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