  We consider the algebra M_k(C) of k-by-k matrices over the complex numbers
and view it as a crossed product with a group G of order k by embedding G in
the symmetric group S_k via the regular representation and embedding S_k in
M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we
call a crossed product grading. This grading is the so called elementary
grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in
G. We study the graded polynomial identities for M_k(C) equipped with a crossed
product grading. To each multilinear monomial in the free graded algebra we
associate a directed labeled graph. This approach allows us to give new proofs
of known results of Bahturin and Drensky on the generators of the T-ideal of
identities and the Amitsur-Levitsky Theorem. Our most substantial new result is
the determination of the asymptotic formula for the G-graded codimension of
M_k(C).
