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

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

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 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

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

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $4$ .

$- 4 - 3 \cdot 2$

Multiply $- 3$ by $2$ .

$- 4 - 6$

Subtract $6$ from $- 4$ .

$- 10$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{- 10} [\begin{array}{c} 4 & - 2 \\ - 3 & - 1 \end{array}]$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{10} [\begin{array}{c} 4 & - 2 \\ - 3 & - 1 \end{array}]$

Step 5

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

$[\begin{array}{c} - \frac{1}{10} \cdot 4 & - \frac{1}{10} \cdot - 2 \\ - \frac{1}{10} \cdot - 3 & - \frac{1}{10} \cdot - 1 \end{array}]$

Step 6

Simplify each element in the matrix.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{10}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{10} \cdot 4 & - \frac{1}{10} \cdot - 2 \\ - \frac{1}{10} \cdot - 3 & - \frac{1}{10} \cdot - 1 \end{array}]$

Factor $2$ out of $10$ .

$[\begin{array}{c} \frac{- 1}{2 (5)} \cdot 4 & - \frac{1}{10} \cdot - 2 \\ - \frac{1}{10} \cdot - 3 & - \frac{1}{10} \cdot - 1 \end{array}]$

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{- 1}{5}$ and $2$ .

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

Multiply $- 1$ by $2$ .

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

Move the negative in front of the fraction.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{10}$ into the numerator.

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

Factor $2$ out of $10$ .

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{- 1}{5}$ and $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $- \frac{1}{10} \cdot - 3$ .

Multiply $- 3$ by $- 1$ .

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

Combine $3$ and $\frac{1}{10}$ .

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{1}{10}$ by $1$ .

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

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

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