Finding the Inverse of the Matrix, If It Exists

We will use the method of augmenting with the identity.

1. Switch row 1 and row 2.
   $$\left[\begin{array}{cc}1& 3\\ 3& 2\end{array}|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$
2. Multiply row 1 by −3 and add it to row 2.
   $$\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}|\begin{array}{cc}1& 0\\ \mathrm{-3}& 1\end{array}\right]$$
3. There is nothing further we can do. The zeros in row 2 indicate that this matrix has no inverse.