Let A be a matrix with entries in Z/nZ. (n is not assumed to be prime.) Then the size of the row span is the size of the column span. All computations are mod n, so both these numbers are finite.
I believe you can prove this using Smith normal form: both the size of the row span and the size of the column span will be the same after passing to the Smith normal form (lift the matrix entries to Z to compute the Smith normal form). When A is in Smith normal form, these quantities can be easily computed.
I would like to find a reference for this fact, and would be grateful to anyone who could provide one.