The following matrix is an augmented matrix
for the system \(Ax = b\) over \(GF(2)\).
\(\begin{bmatrix}
1 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 \\
\end{bmatrix}\)
How many solutions does the system have?
The answer is 4.
First of all, there is no row of 0's in the columns corresponding
to the variables and a nonzero in the right-hand side column.
So the system has at least one solution.
Since there are two free variables, \(x_2\) and \(x_4\), and each
variable can take on the value 0 or 1, there
are a total of \(2 \times 2 = 4\) solutions.