  A tournament is a directed graph resulting from an orientation of the
complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$
is a matrix with $0$s on its diagonal and all other entries equal to $1$. An
outstanding question in tournament theory asks to classify the adjacency
matrices of tournaments which are singular (or nonsingular). We study this
question using the structure of tournaments as graphs, in particular their
cycle structure. More specifically, we find, as precisely as possible, the
number of cycles of length three that dictates whether the corresponding
tournament matrix is singular or nonsingular. We also give structural
classifications of the tournaments that have the specified numbers of cycles of
length three.
