Sufficient condition for the inverse of a matrix

Question

Let $A$ be a square matrix. Another square matrix $B$ is called inverse of $A$ if $$AB=BA=I.$$ My question is whether just $AB=I$ or $BA=I$ is not sufficient to call $B$ as the inverse of $A$? If it is not, then give a counter example where $AB=I$, but $AB\ne BA.$

Answer 1

$AB=I\implies \det A\neq 0\implies A$ is invertible.

$\implies B = A^{-1}$

$\implies BA = I$

Answer 2

Since invertible square matrices over a field $K$ form a group, namely $GL_n(K)$, the left-inverse is equal to the right-inverse - see this duplicate.