  Let G be either of Mat(n), GL(n) or SL(n), let O_q(G) be the quantum function
algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the
specialisation of O_q(G) at a root of unity, of odd order l. Then O_e(G) is a
module over the corresponding classical function algebra O(G) via the quantum
Frobenius morphism, which embeds O(G) as a central subbialgebra of O_e(G).
  In this note we prove a PBW-like theorem for O_q(G) - more or less known in
literature, but not in this form (to the best of the author's knowledge) - and
we show that it yields explicit bases of O_e(G) over O(G) when G is Mat(n) or
GL(n): in particular, O_e(G) is free of rank l^{dim(G)}. Also, we apply the
latter result to prove that O_e(G) is a free Frobenius extensions over O(G),
and to compute explicitly the corresponding Nakayama automorphism, again for G
being Mat(n) or GL(n) . This extends previous results by Brown, Gordon and
Stroppel (see [BG], [BGS2]).
