I'm having difficulty applying modular arithmetic in solving basic systems over finite fields. If I think of it in terms of time, the arithmetic makes a little more sense. For example, means that in right?

Suppose I want to solve the system in that has the augmented matrix:

So whenever I perform a row-op, I have to apply the definitions of modular addition and multiplication on each element. So I could right the matrix obtained by the row-op

This is where I get confused because isn't an element of . So do I just use the fact that , which means that in ? This would give .

Is my reasoning correct so far?

Thanks