Solution: We will use the row-reduce to find the rank of the given matrix. \begin{align*} \begin{bmatrix} 1 & -2 & 2 & 3 & -6 \\ 0 & -1 & -3 & 1 & 1 \\ -2 & 4 & -3 & -6 & 11 \\ \end{bmatrix} \xrightarrow{R_3 \to R_3 + 2R_1} \begin{bmatrix} 1 & -2 & 2 & 3 & -6 \\ 0 & -1 & -3 & 1 & 1 \\ 0 & 0 & 1 & 0 & -1 \\ \end{bmatrix}. \end{align*} Since there are three independent rows of the matrix, and hence the rank of the given matrix will be $3$. Now we will apply the rank-nullity theorem to determine the nullity of the matrix. \[ \operatorname{nullity}(A) = 5 - \operatorname{rank}(A) = 5 - 3 = 2. \]