Answer
$A^{-1}=\left[\begin{array}{cc}0 & 1 \\ -1 / 2 & 3 / 2\end{array}\right]$
Work Step by Step
Given $A=\left[\begin{array}{cc} 3 & -2\\ 1 & 0 \end{array}\right]$
The Matrix Operation is $\quad R_{1} \leftrightarrow R_{2} \quad $
to get $ \left[\begin{array}{cc}1 & 0 \\ 3 & -2\end{array}\right] $
the Elementary matrix
$ E_{1}=\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right]$
The Matrix Operation is $ \quad R_{2}-3 R_{1} \rightarrow R_{2} \quad$ to get $\left[\begin{array}{cc}1 & 0 \\ 0 & -2\end{array}\right] $
the Elementary matrix $E_{2}=\left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]$
The Matrix Operation is $ \quad \frac{-R_{2}}{2} \rightarrow R_{2} \quad$ to get $\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
the Elementary matrix is
$E_{3}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1 / 2\end{array}\right]$
$\therefore A^{-1}=E_{3} E_{2} E_{1}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1 / 2\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -3 & 1\end{array}\right]\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{cc}0 & 1 \\ -1 / 2 & 3 / 2\end{array}\right]$
