## Tournament Matrix

A matrix for a round-robin tournament involving players competing in matches (no ties allowed) having entries

The Matrix satisfies

where is the Identity Matrix, is an Matrix of all 1s, and is the Matrix Transpose of A.

The tournament matrix for players has zero Determinant Iff is Odd (McCarthy and Benjamin 1996). The dimension of the Nullspace of an -player tournament matrix is

(McCarthy 1996).

References

McCarthy, C. A. and Benjamin, A. T. Determinants of the Tournaments.'' Math. Mag. 69, 133-135, 1996.

Michael, T. S. The Ranks of Tournament Matrices.'' Amer. Math. Monthly 102, 637-639, 1995.