State and prove Hall's theorem about matchings in bipartite graphs.

Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix, with all entries non-negative reals, such that every row sum and every column sum is 1. By applying Hall's theorem, show that there is a permutation $\sigma$ of $\{1, \ldots, n\}$ such that $a_{i \sigma(i)}>0$ for all $i$.