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 $- 7$ by $- 2$ .

$- 15 + 14$

Add $- 15$ and $14$ .

$- 1$

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 $- 1$ .

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

Step 5

Multiply $- 1$ by each element of the matrix.

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

Step 6

Simplify each element in the matrix.

Multiply $- 1$ by $- 3$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $- 1$ by $- 7$ .

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

Multiply $- 1$ by $5$ .

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

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

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