# How do you find the inverse of A=((3, 4), (5, 6))?

Feb 13, 2016

Use the general formula for the inverse of a $2 \times 2$ matrix to find:

${\left(\begin{matrix}3 & 4 \\ 5 & 6\end{matrix}\right)}^{- 1} = \left(\begin{matrix}- 3 & 2 \\ \frac{5}{2} & - \frac{3}{2}\end{matrix}\right)$

#### Explanation:

The inverse of a $2 \times 2$ matrix is given by the formula:

${\left(\begin{matrix}a & b \\ c & d\end{matrix}\right)}^{- 1} = \frac{1}{\left\mid \begin{matrix}a & b \\ c & d\end{matrix} \right\mid} \left(\begin{matrix}d & - b \\ - c & a\end{matrix}\right)$

where the determinant is given by the formula:

$\left\mid \begin{matrix}a & b \\ c & d\end{matrix} \right\mid = a d - b c$

In our case $\left(\begin{matrix}a & b \\ c & d\end{matrix}\right) = \left(\begin{matrix}3 & 4 \\ 5 & 6\end{matrix}\right)$ and we find:

$\left\mid \begin{matrix}a & b \\ c & d\end{matrix} \right\mid = \left\mid \begin{matrix}3 & 4 \\ 5 & 6\end{matrix} \right\mid = \left(3 \cdot 6\right) - \left(4 \cdot 5\right) = - 2$

So:

${\left(\begin{matrix}3 & 4 \\ 5 & 6\end{matrix}\right)}^{- 1} = \frac{1}{- 2} \left(\begin{matrix}6 & - 4 \\ - 5 & 3\end{matrix}\right) = \left(\begin{matrix}- 3 & 2 \\ \frac{5}{2} & - \frac{3}{2}\end{matrix}\right)$