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, $\displaystyle 23(\bmod 12)=11$ means that $\displaystyle 23=11$ in $\displaystyle Z_{12}$ right?

Suppose I want to solve the system in $\displaystyle Z_2={0,1}$ that has the augmented matrix:

$\displaystyle \left(\begin{array}{cccc}1&1&1&0\\1&0&1&1\end{arra y}\right)$

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 $\displaystyle R_1\rightarrow R_1+R_2$

$\displaystyle \left(\begin{array}{cccc}1\oplus 1 &1\oplus 0 &1\oplus 1&0\oplus 1\\1&0&1&1\end{array}\right)$

$\displaystyle =\left(\begin{array}{cccc}2(\bmod 2) &1(\bmod 2) & 2(\bmod 2)&1(\bmod 2)\\1&0&1&1\end{array}\right)$

$\displaystyle =\left(\begin{array}{cccc}0&-1&0&-1\\1&0&1&1\end{array}\right)$

This is where I get confused because $\displaystyle -1$ isn't an element of $\displaystyle Z_2$. So do I just use the fact that $\displaystyle 1=-1(\bmod 2)$, which means that $\displaystyle -1=1$ in $\displaystyle Z_2$? This would give $\displaystyle y=1$.

Is my reasoning correct so far?

Thanks