How to Apply Hall's theorem in this question?

Question

A table with $m$ rows and $n$ columns is filled with nonnegative integers such that each row and each column contains at least one positive integer. Moreover, if a row and a column intersect in a positive integer, then the sums of their integers are the same. Using Hall’s theorem or otherwise, prove that $m = n$

Answer 1

For simplicity assume that each entry of the matrix is either $1$ or $0$.

Define a bipartite graph $G=(X,Y)$ where $X=\{r_1,\ldots,r_m\}$ and $Y=\{c_1,\ldots,c_n\}$.

Join $r_i$ with $c_j$ by an edge of and only if the $i,j$-th entry of the marix is $1$.

Note that $\deg(r_i)$ is same as the sum of the entries in the $i$-th row.

Similarly, $\deg(c_j)$ is same the the sum of entries in the $j$-th column.

By hypothesis, $\deg(r_i)=\deg(c_j)$ if $r_ic_j\in E(G)$. Therefore, if there is a path from a vertex $a$ of $G$ to a vertex $b$ of $G$, we must have $\deg(a)=\deg(b)$.

This means that every component of $G$ is a regular bipartite graph.

Hence, by Hall's Theorem, every component of $G$ admits a perfect matching and therefore $|X|=|Y|$.