The Maximal Determinant and Subdeterminants of ±1 Matrices

In this paper we study the maximal absolute values of determinants and subdeterminants of ±1 matrices, especially Hadamard matrices. It is conjectured that the determinants of ±1 matrices of order n can have only the values k • p, where p is specified from an appropriate procedure. This conjecture is verified for small values of n. The question of what principal minors can occur in a completely pivoted ±1 matrix is also studied. An algorithm to compute the (n — j) x (n — j), j = 1, 2, ... minors of Hadamard matrices of order n is presented, and these minors are determined for j =1,...,4.