transformation between rectangular matrices

Question

Given two matrices $A$, $B$ of dimension $m\times n$ where $m<n$, does there always exist a $C$ such that $AC=B$? So by question, the dimension of $C$ is $n \times n$ and a choice of $C=A^R B$, where $A^R$ is the right inverse of $A$. But I am not able to show why $C$ should be invertible. Because of an inverse exists than $C^{-1}C=C^{-1}C=Id$, but I'm not able to find such a $C$ trivially. Is there any nice resource other than Wikipedia that talks about inverses of rectangular matrices in general?

Answer 1

For your first question, what if $A=0$ but $B\ne0$? By the way, if $C$ exists, it is not always invertible --- just pick any $A$ and any non-invertible $C$ and define $B=AC$.

For your second question, see Moore-Penrose pseudoinverse.