Find the Inverse [[5,-2],[-7,3]]

$[\begin{array}{c} 5 & - 2 \\ - 7 & 3 \end{array}]$

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

$5 \cdot 3 - (- 7 \cdot - 2)$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $3$ .

$15 - (- 7 \cdot - 2)$

Multiply $- (- 7 \cdot - 2)$ .

Multiply $- 7$ by $- 2$ .

$15 - 1 \cdot 14$

Multiply $- 1$ by $14$ .

$15 - 14$

Subtract $14$ from $15$ .

$1$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{1} [\begin{array}{c} 3 & 2 \\ 7 & 5 \end{array}]$

Step 4

Divide $1$ by $1$ .

$1 [\begin{array}{c} 3 & 2 \\ 7 & 5 \end{array}]$

Step 5

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 1 \cdot 3 & 1 \cdot 2 \\ 1 \cdot 7 & 1 \cdot 5 \end{array}]$

Step 6

Simplify each element in the matrix.

Multiply $3$ by $1$ .

$[\begin{array}{c} 3 & 1 \cdot 2 \\ 1 \cdot 7 & 1 \cdot 5 \end{array}]$

Multiply $2$ by $1$ .

$[\begin{array}{c} 3 & 2 \\ 1 \cdot 7 & 1 \cdot 5 \end{array}]$

Multiply $7$ by $1$ .

$[\begin{array}{c} 3 & 2 \\ 7 & 1 \cdot 5 \end{array}]$

Multiply $5$ by $1$ .

$[\begin{array}{c} 3 & 2 \\ 7 & 5 \end{array}]$

Find the Inverse [[5,-2],[-7,3]]

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