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

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

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

$5 \cdot 5 - 2 \cdot 2$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $5$ .

$25 - 2 \cdot 2$

Multiply $- 2$ by $2$ .

$25 - 4$

Subtract $4$ from $25$ .

$21$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{21} [\begin{array}{c} 5 & - 2 \\ - 2 & 5 \end{array}]$

Step 4

Multiply $\frac{1}{21}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{21} \cdot 5 & \frac{1}{21} \cdot - 2 \\ \frac{1}{21} \cdot - 2 & \frac{1}{21} \cdot 5 \end{array}]$

Step 5

Simplify each element in the matrix.

Combine $\frac{1}{21}$ and $5$ .

$[\begin{array}{c} \frac{5}{21} & \frac{1}{21} \cdot - 2 \\ \frac{1}{21} \cdot - 2 & \frac{1}{21} \cdot 5 \end{array}]$

Combine $\frac{1}{21}$ and $- 2$ .

$[\begin{array}{c} \frac{5}{21} & \frac{- 2}{21} \\ \frac{1}{21} \cdot - 2 & \frac{1}{21} \cdot 5 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{21} & - \frac{2}{21} \\ \frac{1}{21} \cdot - 2 & \frac{1}{21} \cdot 5 \end{array}]$

Combine $\frac{1}{21}$ and $- 2$ .

$[\begin{array}{c} \frac{5}{21} & - \frac{2}{21} \\ \frac{- 2}{21} & \frac{1}{21} \cdot 5 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{21} & - \frac{2}{21} \\ - \frac{2}{21} & \frac{1}{21} \cdot 5 \end{array}]$

Combine $\frac{1}{21}$ and $5$ .

$[\begin{array}{c} \frac{5}{21} & - \frac{2}{21} \\ - \frac{2}{21} & \frac{5}{21} \end{array}]$

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

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