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

TL;DR

This paper investigates properties of 0/1 matrices with maximum determinants, establishing necessary conditions involving inverse matrix row sums, and characterizes when these determinants are maximal.

Contribution

It provides new necessary conditions for 0/1 matrices to have maximum determinants, linking inverse matrix properties to maximality.

Findings

Necessary conditions involving inverse matrix row sums for maximal determinants.

Characterization of when the determinant of a 0/1-matrix is maximal.

Insights into the structure of matrices achieving maximum determinants.

Abstract

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