Find the inverse of each of the matrices, if it exists.

Let $A=\left[\begin{array}{ll}3 & 1 \\ 5 & 2\end{array}\right]$

We know that A = AI

$\therefore\left[\begin{array}{ll}3 & 1 \\ 5 & 2\end{array}\right]=A\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]=A\left[\begin{array}{ll}1 & 0 \\ -2 & 1\end{array}\right] \quad\left(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-2 \mathrm{C}_{2}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]=A\left[\begin{array}{lr}1 & -1 \\ -2 & 3\end{array}\right] \quad\left(\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=A\left[\begin{array}{lr}2 & -1 \\ -5 & 3\end{array}\right] \quad\left(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}\right)$

$\therefore A^{-1}=\left[\begin{array}{lr}2 & -1 \\ -5 & 3\end{array}\right]$