Relationship between number of perfect matchings in a balanced bipartite graph and determinant of its biadjacency matrix.

Question

Let $G$ be a balanced bipartite graph with the adjacency matrix $A$. It is known that $A$ can be written in the form $A=\left[\begin{array}{ll} 0&B\\ B^T&0 \end{array}\right]$, where $B$ is termed as the biadjacency matrix of $G$. I wanted know whether we can say any thing about number of perfect matchings in a bipartite graph given the determinant of its biadjacency matrix. From Godsil's "Inverses of trees" (Combinatorica, 5(1), 1985), Lemma 2.1 we have that "$G$ has a unique perfect matching if and only if $B$ can be written as lower triangular matrix with all its diagonal entries equal to 1". Clearly, from this we can deduce that if $G$ has unique perfect matching then $|\det B|=1$. My question: Does the converse hold? If no, under what condition(s) we can ensure that the converse hold.