Solution: We will use the row reduction to find out the rank and nullity of the given matrix. \begin{align*} & \begin{pmatrix} 0 & 0 & -1 & 5 \\ 0 & 0 & -3 & 8 \\ 0 & 0 & 1 & 2 \\ \end{pmatrix} \xrightarrow{R_1 \rightarrow -R_1} \begin{pmatrix} 0 & 0 & 1 & -5 \\ 0 & 0 & -3 & 8 \\ 0 & 0 & 1 & 2 \\ \end{pmatrix} \\[1ex] \xrightarrow[R_3 \rightarrow R_3-R_1]{R_2 \rightarrow R_2+3R_1} & \begin{pmatrix} 0 & 0 & 1 & -5 \\ 0 & 0 & 0 & -7 \\ 0 & 0 & 0 & 7 \\ \end{pmatrix} \xrightarrow{R_3 \rightarrow R_3 + R_2} \begin{pmatrix} 0 & 0 & 1 & -5 \\ 0 & 0 & 0 & -7 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}. \end{align*} Since the matrix has two non-zero rows and one zero row, the rank of the matrix is $2$ and the nullity is one.