Domino tilings and determinants

Consider an arbitrary simply connected squared figure $F$ on the plane and its dual graph (vertices correspond to cells, edges correspond to cells sharing a common side). We investigate the relationship between the determinant of the adjacency matrix of the graph and the domino tilings of the figure $F$. We prove that in the case where all the tilings can be splitted into pairs such that the numbers of vertical dominos in each pair differ by 1, then $\operatorname{det}A_F=0$. And in the case where all the tilings except one can be splitted into such pairs, $\operatorname{det}A_F=(-1)^s$, where $s$ is half the area of the figure $F$.