I am studying from the book and there is an example given which I can't figure out.
Solving a matrix in $\Bbb Z_p$ where p is a prime number has many respects like $\Bbb R$; like adding, substracting, multiplying and dividing.
If you have the equation $x_1 + x_2 + x_3 = 1$ in $\Bbb Z_2$ there are 4 solutions given.
The solutions which are given are:
$\begin{bmatrix}x\\y\\z\end{bmatrix}$=$\begin{bmatrix}1\\0\\0\end{bmatrix}$,$\begin{bmatrix}0\\1\\0\end{bmatrix}$,$\begin{bmatrix}0\\0\\1\end{bmatrix}$,$\begin{bmatrix}1\\1\\1\end{bmatrix}$
As you see gives the last solution:
$x_1 + x_2 + x_3 = 1$
$1 + 1 + 1 = 1$
The first three seems logic to me, but I am not sure why the forth solution is legit.
In $\mathbb{Z}_2$, $1+1=0$ and therefore $1+1+1=0+1=1$.