Find the Inverse [[9,-5],[7,-4]]

$[\begin{array}{c} 9 & - 5 \\ 7 & - 4 \end{array}]$

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $| A |$ is the determinant of $A$ .

If $A = [\begin{array}{c} a & b \\ c & d \end{array}]$ then $A^{- 1} = \frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$

Find the determinant of $[\begin{array}{c} 9 & - 5 \\ 7 & - 4 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} 9 & - 5 \\ 7 & - 4 \end{array}] = | \begin{array}{c} 9 & - 5 \\ 7 & - 4 \end{array} |$

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$(9) (- 4) - 7 \cdot - 5$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 4$ .

$- 36 - 7 \cdot - 5$

Multiply $- 7$ by $- 5$ .

$- 36 + 35$

Add $- 36$ and $35$ .

$- 1$

Substitute the known values into the formula for the inverse of a matrix.

$\frac{1}{- 1} [\begin{array}{c} - 4 & - (- 5) \\ - (7) & 9 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 5)$ .

$\frac{1}{- 1} [\begin{array}{c} - 4 & 5 \\ - (7) & 9 \end{array}]$

Rearrange $- (7)$ .

$\frac{1}{- 1} [\begin{array}{c} - 4 & 5 \\ - 7 & 9 \end{array}]$

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

$[\begin{array}{c} \frac{1}{- 1} \cdot - 4 & \frac{1}{- 1} \cdot 5 \\ \frac{1}{- 1} \cdot - 7 & \frac{1}{- 1} \cdot 9 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{- 1} \cdot - 4$ .

$[\begin{array}{c} 4 & \frac{1}{- 1} \cdot 5 \\ \frac{1}{- 1} \cdot - 7 & \frac{1}{- 1} \cdot 9 \end{array}]$

Rearrange $\frac{1}{- 1} \cdot 5$ .

$[\begin{array}{c} 4 & - 5 \\ \frac{1}{- 1} \cdot - 7 & \frac{1}{- 1} \cdot 9 \end{array}]$

Rearrange $\frac{1}{- 1} \cdot - 7$ .

$[\begin{array}{c} 4 & - 5 \\ 7 & \frac{1}{- 1} \cdot 9 \end{array}]$

Rearrange $\frac{1}{- 1} \cdot 9$ .

$[\begin{array}{c} 4 & - 5 \\ 7 & - 9 \end{array}]$

Find the Inverse [[9,-5],[7,-4]]

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