Is it possible that $A\cdot B= I$, but that $B\cdot A\neq I$?

Question

I know that $A\cdot A^{-1}=A^{-1}\cdot A = I$, but is it possible for a matrix $B$ to exist such that $A\cdot B= I$, but $B\cdot A\neq I$? If that it is not the case, why not?

($I$ is the identity matrix)

Answer 1

Yes, if you don’t require the matrices to be square; J. W. Tanner’s answer provides an example. If you do require $B$ to be square, however, it is true that if $AB=I$, then $BA=I$; there is a short proof here, for instance.

Answer 2

It's possible if $A$ and $B$ are not square matrices;

for example, $\pmatrix{1&0}\pmatrix{1\\0}=\pmatrix{1}$, but $\pmatrix{1\\0}\pmatrix{1&0}=\pmatrix{1&0\\0&0}$