Find the Inverse [[2,4],[5,1]]

$[\begin{array}{c} 2 & 4 \\ 5 & 1 \end{array}]$

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

$2 \cdot 1 - 5 \cdot 4$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $1$ .

$2 - 5 \cdot 4$

Multiply $- 5$ by $4$ .

$2 - 20$

Subtract $20$ from $2$ .

$- 18$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{- 18} [\begin{array}{c} 1 & - 4 \\ - 5 & 2 \end{array}]$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{18} [\begin{array}{c} 1 & - 4 \\ - 5 & 2 \end{array}]$

Step 5

Multiply $- \frac{1}{18}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{18} \cdot 1 & - \frac{1}{18} \cdot - 4 \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Step 6

Simplify each element in the matrix.

Multiply $- 1$ by $1$ .

$[\begin{array}{c} - \frac{1}{18} & - \frac{1}{18} \cdot - 4 \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} - \frac{1}{18} & \frac{- 1}{18} \cdot - 4 \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Factor $2$ out of $18$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{- 1}{2 (9)} \cdot - 4 \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Factor $2$ out of $- 4$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{- 1}{2 \cdot 9} \cdot (2 \cdot - 2) \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{18} & \frac{- 1}{2 \cdot 9} \cdot (2 \cdot - 2) \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{18} & \frac{- 1}{9} \cdot - 2 \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Combine $\frac{- 1}{9}$ and $- 2$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{- - 2}{9} \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ - \frac{1}{18} \cdot - 5 & - \frac{1}{18} \cdot 2 \end{array}]$

Multiply $- \frac{1}{18} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ 5 (\frac{1}{18}) & - \frac{1}{18} \cdot 2 \end{array}]$

Combine $5$ and $\frac{1}{18}$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ \frac{5}{18} & - \frac{1}{18} \cdot 2 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ \frac{5}{18} & \frac{- 1}{18} \cdot 2 \end{array}]$

Factor $2$ out of $18$ .

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ \frac{5}{18} & \frac{- 1}{2 (9)} \cdot 2 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ \frac{5}{18} & \frac{- 1}{2 \cdot 9} \cdot 2 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ \frac{5}{18} & \frac{- 1}{9} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{18} & \frac{2}{9} \\ \frac{5}{18} & - \frac{1}{9} \end{array}]$

Find the Inverse [[2,4],[5,1]]

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