  A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that
contains all the vertices. Two perfect matchings intersect if they share an
edge. It is known that if $\mathcal{F}$ is family of intersecting perfect
matchings of $K_{2n}$, then $|\mathcal{F}| \leq (2(n-1) - 1)!!$ and if equality
holds, then $\mathcal{F} = \mathcal{F}_{ij}$ where $ \mathcal{F}_{ij}$ is the
family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$.
We give a short algebraic proof of this result, resolving a question of Godsil
and Meagher. Along the way, we show that if a family $\mathcal{F}$ is
non-Hamiltonian, that is, $m \cup m' \not \cong C_{2n}$ for any $m,m' \in
\mathcal{F}$, then $|\mathcal{F}| \leq (2(n-1) - 1)!!$ and this bound is met
with equality if and only if $\mathcal{F} = \mathcal{F}_{ij}$. Our results make
ample use of a somewhat understudied symmetric commutative association scheme
arising from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. We give an exposition of
a few new interesting objects that live in this scheme as they pertain to our
results.
