The Determinant of $\{\pm 1\}$-Matrices and Oriented Hypergraphs.

We introduce a set of highly symmetric families that each calculate the determinant of a given $\{\pm 1\}$-matrix by using the oriented hypergraphic Laplacian and the incidence-based notion of cycle-covers. Any non-edge-monic family of cycle-covers is shown to vanish in every determinant, while any one of the $n!$ remaining edge-monic families is equivalent to determining the absolute value of the determinant. Hadamard's maximum determinant problem is shown to be equivalent to optimizing the number of locally signed graphic circles of a given sign in any one of these families or across all of these families. Various symmetries regarding orthogonality, equivalence to $\{0,+1\}$-matrices, and their relation to these edge-monic families are shown to be different fundamental circles related by theta-subgraphs.