  We consider certain functional identities on the matrix algebra $M_n$ that
are defined similarly as the trace identities, except that the "coefficients"
are arbitrary polynomials, not necessarily those expressible by the traces. The
main issue is the question of whether such an identity is a consequence of the
Cayley-Hamilton identity. We show that the answer is affirmative in several
special cases, and, moreover, for every such an identity $P$ and every central
polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that
the affirmative answer holds for $c^mP$. In general, however, the answer is
negative. We prove that there exist antisymmetric identities that do not follow
from the Cayley-Hamilton identity, and give a complete description of a certain
family of such identities.
