  In this paper we study algebras with trace and their trace polynomial
identities over a field of characteristic 0. We consider two commutative matrix
algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the
algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We
describe all possible traces on these algebras and we study the corresponding
trace codimensions. Moreover we characterize the varieties with trace of
polynomial growth generated by a finite dimensional algebra. As a consequence,
we see that the growth of a variety with trace is either polynomial or
exponential.
