Why can't this matrix have a right inverse? 
Let $A$ be an $m \times n$ matrix with m > n. Why can't $A$ have a right inverse.

We want $AB = I_m$, why is this impossible if $m > n$? 

Think about $A$ as the linear transformation that it represents: $T:\mathbf{R}^n\to \mathbf{R}^m$. We know that the number of columns corresponds to the dimension of the domain, while the number of rows corresponds to the dimension of the codomain. We know, then, that $m>n$. So, we know that $T$ can not be surjective. Existence of a right inverse is equivalent to surjectivity. Therefore, $T$ has no right inverse. Similarly, $A$ has no right inverse. 
