No, permutation matrix would be a special case where m=n and all the numbers ai and bj are equal to 1. (Existence in that case would be trivial, since the identity matrix works.) Try to prove this as a special case of the integrality theorem for networks with supplies and arc capacities: If all supplies are integral and all arc capacities are integral, then if there exists a feasible flow then there exists a feasible integral flow. This is somewhat challenging, and most students have not succeeded in doing it. It depends on setting up the right network with the right arc capacities, and showing that it has a feasible flow. You must also show that the integral feasible flows correspond to the matrices you are looking for.

The closest thing we did to this in lecture was in part of the proof of the Birkhoff Von Neumann theorem when we showed that in any nonnegative nxn square matrix where the rows and columns all add up to the same sum it is possible to find a collection of n positive entries all in different rows and columns.

The marriage algorithm, described in http://www.math.gatech.edu/~spingarn/4580/summaries/transship2.pdf can be used to find a maximum matching in a bipartite graph as follows. Think of the nodes on the left as "boys" and the nodes on the right as "girls". A matching is a set of marriages. You keep marrying boys until it is no longer possible to do so. You start by locating an unmarried boy (a node on the left that does not belong to an arc in the current matching). Then you look at all the girls he knows (all the nodes on the right that are joined to him by an arc). If one of them is unmarried (does not belong to an arc in the current matching), then marry her to him, and so forth, as described in the constructive marriage algorithm.