Why only either of $A^{-1}A$ and $AA^{-1}$ is equal to identity matrix? Consider $6\times 4$ matrix $A$. 
Why either of $A^{-1}A$ and $AA^{-1}$ is equal to identity matrix, but the other one is not equal to identity matrix? 
Note: I try to do it numerically as follows. But I don’t understand why I get such a result. I see it’s mathematical explanation in order to convenience. 


Your matrix $A$ is injective but not surjective. Therefore there exists a left inverse for $A$ which is a $4 \times 6$ matrix $B$ such that $$BA = I_{4}$$
If we also had $$AB = I_{6}$$
this would imply that $A$ is surjective, which is false.
