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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
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