linear algebra - Inverse of $2times2$ matrix

Let $A$ and $B$ be two $2\times2$ square matrices such that $AB=I$ Then prove that $BA=I$.

I tried to do it by defining $A$ and $B$ manually by listing the elements but calculations become tedious.
Is there a short way to obtain this result?

One is only allowed to use concept of product of matrices and elementry transformations.