Products of quantum matrices

Given two quantum matrices A is an element of GL(q)(n) and A' is an element of GL(q)' (n) can one impose quadratic relations between elements of A and A' which (a) are consistent and (b) imply that the product matrix AA' is i n GL(q)"(n)? This question is studied systematically in two and three dimen sions. In two dimensions, five families of solutions are found, one of whic h generalizes the relations between A and its inverse A(-1). For each famil y, the results are compared with the quantum transpose condition introduced by Ogievetsky and Wess. In three dimensions, we prove that non-trivial sol utions are absent, a fact which prevents further generalization.