I tried to prove that if $A$ and $B$ are both $n\times n$ matrices and if $AB = I_n$ then $BA = I_n$ (i.e. the matrix $A$ is invertible). So first I managed to conclude that if exists both $B$ and $C$ such that $AB = I_n$ and $CA = I_n$, then trivially $B=C$ . However to conclude the proof we need to show that if such a right inverse exists, then a left inverse must exist too.
No idea how to proceed. All I can use is definition of matrices, and matrix multiplication, sum , transpose and rank.
(I saw proof of this in other questions, but they used things like determinants or vectorial spaces, but I need a proof without that).