Properties of 0/1-Matrices of Order n Having Maximum Determinant

Abstract

We give some necessary conditions for maximality of $0/1$-determinant. Let $\mathbf{M}$ be a nondegenerate $0/1$-matrix of order $n$. Denote by $\mathbf{A}$ the matrix of order $n+1$ which appears from $\mathbf{M}$ after adding the $(n+1)$th row $(0,0,\dots ,0,1)$ and the $(n+1)$th column consisting of $1$'s. Suppose ${\mathbf{A}}^{-1}=(l_{ij}),$ then for all $i=1,\dots ,n$ we have ${\sum }_{j=1}^{n+1}|l_{ij}|\ge 2.$ Moreover, if $|\det (\mathbf{M})|$ is equal to the maximum value of a $0/1$-determinant of order $n$, then ${\sum }_{j=1}^{n+1}|l_{ij}|=2$ for all $i=1,\dots ,n$. Keywords: maximum 0/1-deteminant, simplex, cube, axial diameter

11 pages, 2 figures