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

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

We know that A = IA

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

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

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

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

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