  There are various results connecting ranks of incidence matrices of graphs
and hypergraphs with their combinatorial structure. Here, we consider the
generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for
one natural class of hypergraphs: the triple systems with index three. Such
systems with nonsingular $N_2$ (over the rationals) appear to be quite rare,
yet they can be constructed with PBD closure. In fact, a range of ranks near
$\binom{v}{2}$ is obtained for large orders $v$.
