  In this paper we give a proof that the largest set of perfect matchings, in
which any two contain a common edge, is the set of all perfect matchings that
contain a fixed edge. This is a version of the famous Erd\H{o}s-Ko-Rado theorem
for perfect matchings. The proof given in this paper is algebraic, we first
determine the least eigenvalue of the perfect matching derangement graph and
use properties of the perfect matching polytope. We also prove that the perfect
matching derangement graph is not a Cayley graph.
