I encountered an exercise (not my homework)
Let $A,B,C$ be matrices and $$ M=\begin{pmatrix} A & C \\ 0 & B \end{pmatrix} $$ then $$ \mathrm{rank}~M\ge \mathrm{rank}~A+\mathrm{rank}~B $$ and the equality holds if and only if there exist matrices $X,Y$ such that $C=AX+YB$.
I can work out the inequality (by proving the existence of a nonvanishing minor of size $ \mathrm{rank}~A+\mathrm{rank}~B$) and the "if" part (by multiplying block elementary matrices). However, I failed to handle the "only if" part. I am aware of a paper by W.Roth that deals with the problem, but the method seems to be tedious and out of my reach. In addition, this reminds me of the Sylvester's Inequality, but I cannot proceed further from that.
I wonder if there is a simpler approach, perhaps in the language of linear transforms.
Thanks in advance.