  Let $W_{k,n}^{i}(m)$ denote a matrix with rows and columns indexed by the
$k$-subsets and $n$-subsets, respectively, of an $m$-element set. The row $S$,
column $T$ entry of $W_{k,n}^{i}(m)$ is $1$ if $|S \cap T| = i$, and is $0$
otherwise. We compute the rank of the matrix $W_{2,n}^{1}(m)$ over any field by
making use of the representation theory of the symmetric group. We also give a
simple condition under which $W_{k,n}^{i}(m)$ has large $p$-rank.
