Hello all,
I'm having trouble with a homework problem and am looking for some guidance.

The problem is as follows:
"Over Z3 select a matrix X such that the set of all linear combinations of 1m and X is a 9-element field."
(The 3 is Z3 and m in 1m are supposed to be sub-scripted)

What I understand so far:
I assume he means a 2x2 matrix from what he said in class.
1m is the identity matrix.
A linear combination is in the form z=u*X+v*1m (Right?)

what I dont understand:

In the linear combination are u and v other matrices or scalars? If they're scalars they're just 0,1,2 right? How can that make a 9-element field regardless of what the matrix X is?

Any help/suggestions/comments would be appreciated.