Summary: The (0, 1)-matrix A of order n is a tournament matrix provided A + A T+ I = J, where I is the identity matrix, and J = J n is the all 1's matrix of order n. It was shown by de Caen and Michael that the rank of a tournament matrix A of order n over a field of characteristic p satisfies rank p (A) $\geq $(n - 1)/2 with equality if and only if n is odd and AA T= O. This article shows that the rank of a tournament matrix A of even order n over a field of characteristic p satisfies rank p (A) $\geq $n/2 with equality if and only if after simultaneous row and column permutations AA $T= \pm J$ m OO O, for a suitable integer m. The results and constructions for even order tournament matrices are related to and shed light on tournament matrices of odd order with minimum rank.

15A03, 05C20, 05C50

tournament matrix, rank