# How do you find the inverse of A=((8, 6), (7, 5))?

Sep 6, 2016

${A}^{-} 1 = \left(\begin{matrix}- 2 \frac{1}{2} & 3 \\ 3 \frac{1}{2} & - 4\end{matrix}\right)$

#### Explanation:

For $A = \left(\begin{matrix}8 & 6 \\ 7 & 5\end{matrix}\right)$

To find the inverse of a 2x2 matrix: $\text{ } X = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$

1. Find the determinant: $\left\mid X \right\mid = a b - c d$
2. Change the matrix to $\left(\begin{matrix}d & - b \\ - c & a\end{matrix}\right)$
3. Divide by the determinant

${A}^{-} 1 = \frac{1}{\left\mid A \right\mid} \left(\begin{matrix}5 & - 6 \\ - 7 & 8\end{matrix}\right)$

$\left\mid A \right\mid = 8 \times 5 - 7 \times 6 = - 2$

${A}^{-} 1 = \frac{1}{-} 2 \left(\begin{matrix}5 & - 6 \\ - 7 & 8\end{matrix}\right)$

${A}^{-} 1 = \left(\begin{matrix}- 2 \frac{1}{2} & 3 \\ 3 \frac{1}{2} & - 4\end{matrix}\right)$