The allowed rules are:
1. Multiply any row (or column) by any non-zero number (so yes, you are right, you can do this).
2. Add (or subtract) a constant non-zero multiple of one row (or column) to another row.
3. Swap two rows over.
I've never done it for a "modulo n" matrix myself, but it's okay, the modulus is prime, so the elements of the matrix are elements of a field.
Not quite sure what order I'd do stuff but if you add row 1 to row 2 then your first element of row 2 ends up being 1, and you can then subtract it from row 3, or something ...