Properties of (0, 1)-matrices of order n having maximal determinant

  • Nevskii Mikhail, mnevsk55@yandex.ru Department of Mathematics, P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150003, Russia
  • Ukhalov Alexey, alex-uhalov@yandex.ru Department of Mathematics, P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150003, Russia
Keywords: (0,1)-matrix with the maximal determinant, simplex, cube, axial diameter

Abstract

We give some necessary conditions for the maximality of $(0, 1)$-determinant. Let $M$ be a nondegenerate $(0, 1)$-matrix of order $n$. Denote by $A$ the matrix of order $n + 1$ which is obtained from $M$ by adding the $(n + 1)$-th row $(0, 0, \dots, 0, 1)$ and the $(n + 1)$-th column consisting of 1’s. We prove that if $A^{−1} = (l_{i,j})$ then for all $i = 1,\dots,n$ we have $\sum\limits^{n+1}_{j=1}|l_{i,j}|\geq2$. Moreover, if $|det(M)|$ is equal to the maximal value of a $(0, 1)$-determinant of order $n$, then $\sum\limits^{n+1}_{j=1}|l_{i,j}|=2$ for all $i = 1,\dots, n$.

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How to Cite
Nevskii, M. and Ukhalov, A. (2019) “Properties of (0, 1)-matrices of order n having maximal determinant”, Mathematical notes of NEFU, 26(2), pp. 109-115. doi: https://doi.org/10.25587/SVFU.2019.102.31516.
Section
Mathematical Modeling