  In this paper we study of *-representations for polynomial algebras on
quantum matrix spaces. We deal with two special cases of the polynomial
algebras, namely the algebra of polynomials on quantum complex matrices
$\mathrm{Mat_2}$ and on quantum complex symmetric matrices
$\mathrm{Mat_2^{sym}}$. For the second algebra we classify all irreducible
*-representations by bounded operators in a Hilbert space (up to a unitary
equivalence). Moreover, we present a construction of *-representations of the
above algebras which enables to obtain the full list of *-representations
(sometimes by passing to subrepresentations).
