linear algebra - Are there matrices such that $AB=I$ and $BA neq I$

Are there matrices such that $AB=I$ and $BA \neq I$ ?

$A$ and $B$ are square matrices

Answer

With square matrices, no. With non-square matrices, it's perfectly possible. For example,
$$\begin{align*} A &= \left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right) \\ B &= \left(\begin{array}{cc}1 & 0 \\ 0 & 1 \\ 0 & 0\end{array}\right) \end{align*}$$
Note that, indeed, we cannot have non-square matrices where $AB$ and $BA$ are identities of appropriate dimensions, because that would imply the existence of an isomorphism between spaces of different dimensions!