If $A$ and $B$ are two matrices, and $AB=I$ (of suitable order) is $BA=I$ always true?

Question

It is said that a matrix $A$ is said to be invertible if there exists a matrix $B$ such that $AB=I=BA$, but if we only know $AB=I$, can we be sure that $BA=I$?

Answer 1

If $A$ is invertible then $$AB=I\implies ABA=IA=A\implies A^{-1}ABA=A^{-1}A\implies BA=I.$$

Answer 2

If your matrix are square, then is true. A proof is here. The proof doesn't suppose $A^{-1}$ exists. If not, for example $$ A = \begin{pmatrix} 2 & 0 & 1\\ 0 & 0 & 2 \end{pmatrix}\qquad B = \frac{1}{4}\begin{pmatrix} 2 & -1\\ 0 & 0 \\ 0 & 2 \end{pmatrix} $$ Then $$ AB= I_2 \qquad BA = \begin{pmatrix}1& 0 &0\\ 0 & 0 &0 \\ 0& 0 & 1\end{pmatrix} $$