  The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented
hypergraphic Laplacian and summing over an incidence generalization of vertex
cycle-covers. These cycle-covers are signed and partitioned into families based
on their hyperedge containment. Every non-edge-monic family is shown to
contribute a net value of $0$ to the Laplacian, while each edge-monic family is
shown to sum to the absolute value of the determinant of the original incidence
matrix. Simple symmetries are identified as well as their relationship to
Hadamard's maximum determinant problem. Finally, the entries of the incidence
matrix are reclaimed using only the signs of an adjacency-minimal set of
cycle-covers from an edge-monic family.
