Example:
\[\mathbf{M} = \left( \begin{array}{ccc} 1 & 2 & 1 \\ 1 & -1 & 0 \\ -1 & -2 & -1 \\ 2 & 1 & 1 \end{array} \right) \]
\[ \left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 3 & -1 \end{array} \right) \begin{array}{c} \\ R2-R1 \\ R3+R1 \\R4-2*R1 \end{array}\]
\[ \left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \begin{array}{c} \\ \\ \\R4-R2 \end{array}\]
There are two non zero rows so the rank of the matrix is 2 and there are two zero rows so the nullity is 2.