Bipartite adjacency matrix

With the two vertex classes listed consecutively, a bipartite graph has adjacency matrix $\left(\begin{array}{cc}0& B\\ B^T& 0\end{array}\right)$.

If a square bipartite adjacency matrix $B$ has nonzero determinant, some product in its Leibniz expansion is nonzero. Its permutation selects one edge at each vertex and therefore gives a perfect matching.