# Determinants of the Tournaments

by Clifford A. McCarthy (Harvey Mudd College) and Arthur T. Benjamin (Harvey Mudd College)

This article originally appeared in:
Mathematics Magazine
April, 1996

Subject classification(s): Algebra and Number Theory | Linear Algebra
Applicable Course(s): 3.8 Linear/Matrix Algebra

Consider a tournament with $n$ players where each player plays every other player once, and ties are not allowed.  An $n \times n$ tournament matrix $A$ is constructed where diagonal entries are zero, $A_{ij} = 1$ if $i$ beats $j$, and $A_{ij}=-1$ if $j$ beats $i$.  The authors demonstrate that the determinant of a tournament matrix is zero if and only if $n$ is odd.  Additionally, it is shown that the nullspace of a tournament matrix has dimension zero if $n$ is even and dimension one if $n$ is odd.

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Capsule Course Topic(s):
Linear Algebra | Determinants
Linear Algebra | Vector Spaces, Subspaces