  We investigate the language classes recognized by group automata over matrix
groups. For the case of $2 \times 2 $ matrices, we prove that the corresponding
group automata for rational matrix groups are more powerful than the
corresponding group automata for integer matrix groups. Finite automata over
some special matrix groups, such as the discrete Heisenberg group and the
Baumslag-Solitar group are also examined. We also introduce the notion of time
complexity for group automata and demonstrate some separations among related
classes. The case of linear-time bounds is examined in detail throughout our
repertory of matrix group automata.
