On 0--1 matrices whose inverses have entries of the same modulus

Abstract

A conjecture of Barrett, Butler and Hall may be stated as follows: If $n \geq 3$ and $A \in \{0,1\}^{n \times n}$ (the family of $n \times n$ 0--1 matrices) is a nonsingular symmetric matrix, then the following two statements are equivalent: (a) All of the principal minors of $A$ of order $n-2$ are zero; and (b) $A^{-1}$ is a matrix all of whose entries have the same modulus and all of whose diagonal entries are equal. We show that this conjecture holds if $A$ does not have both a zero and a nonzero principal minor of order $n-4$ (if $n \geq 5$). The parity of the principal minors of nonsingular symmetric matrices $A \in \{0,1\}^{n \times n}$ whose principal minors of order $n-2$ are all zero is explored, establishing, in particular, that the determinants of such matrices are all even. For an arbitrary (not necessarily symmetric) nonsingular matrix $A \in \{0,1\}^{n \times n}$ with $n\geq 3$, we establish necessary conditions for $A^{-1}$ to be a matrix all of whose entries have the same modulus; examples of such conditions are the following: each row and column of $A$ has an even number of nonzero entries; each entry of $A^{-1}$ is the reciprocal of an even integer; $\det(A)$ is even; the difference between any two rows of $A$, as well as the difference between any two columns of $A$, has an even number of nonzero entries; if $A$ is symmetric, then $A$ has an even number of nonzero diagonal entries; if $A$ is symmetric and $\vec{a}_k$ is the $k$th column of $A$, then $A-\vec{a}_k\vec{a}_k^T$ has an even number of nonzero diagonal entries.