I have several questions about the problem statement.
Do you consider regular matrix multiplication, as in linear algebra, but only in Z[n]?
Could you remind why this it a group? E.g., when one has just numbers, i.e., 1x1-matrices, then Z[n] is not always a group under multiplication, is it?
Finally, how is a group determined by a matrix M? Is M a generating element, i.e., does the group consists of all powers of M? Or do we get different group elements by substituting different numbers for a, b, and c?