  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.
