  Several intersection matrices of $s$-subsets vs. $k$-subsets of a $v$-set are
introduced in the literature. We study these matrices systematically through
counting arguments and generating function techniques. A number of new or known
identities appear as natural consequences of this viewpoint; especially,
appearance of the derivative operator $d/dz$ and some related operators reveals
some connections between intersection matrices and the "combinatorics of
creation-annihilation". As application, the eigenvalues of several intersection
matrices including some generalizations of the adjacency matrices of the
Johnson scheme are derived; two new bases for the Bose--Mesner algebra of the
Johnson scheme are introduced and the associated intersection numbers are
obtained as well. Finally, we determine the rank of some intersection matrices.
