$\underline {\textbf {König - Egerváry Theorem}}$ $:$
Consider a $(0,1)$ matrix $A$. Then the minimum number of lines (either horizontal or vertical) required to cover all the $1$'s of $A$ is equal to the maximum number of $1$'s in $A$ no two of them on a line.
Our instructor has given us a proof of the above theorem by considering a mimimum covering of all the $1$'s of $A$ by $r$ rows and $s$ columns where both of $r$ and $s$ are positive. The proof is relatively simple which uses Hall's theorem. But my question is how can I ensure that neither $r$ nor $s$ is zero? It may so happen that one of them is zero. Then how do I prove the theorem? Please help me in this regard.
Thank you very much.