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(D) Given matrix $A = \begin{bmatrix} 1 & -1 & 0 & -2 \ -4 & 4 & 0 & 8 \ -2 & 1 & 2 & 4 \end{bmatrix}$.We apply elementary row operations to transfo

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$2$

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(D) Given matrix $A = \begin{bmatrix} 1 & -1 & 0 & -2 \ -4 & 4 & 0 & 8 \ -2 & 1 & 2 & 4 \end{bmatrix}$.We apply elementary row operations to transform the matrix into row echelon form.Step $1$: Apply $R_2 ightarrow R_2 + 4R_1$ and $R_3 ightarrow R_3 + 2R_1$.$A \sim \begin{bmatrix} 1 & -1 & 0 & -2 \ 0 & 0 & 0 & 0 \ 0 & -1 & 2 & 0 \end{bmatrix}$.Step $2$: Swap $R_2$ and $R_3$ $(R_2 \leftrightarrow R_3)$.$A \sim \begin{bmatrix} 1 & -1 & 0 & -2 \ 0 & -1 & 2 & 0 \ 0 & 0 & 0 & 0 \end{bmatrix}$.The matrix is now in row echelon form. The number of non-zero rows is $2$.Therefore,the rank of matrix $A$ is $2$.

The rank of the matrix $A=\begin{bmatrix} 1 & -1 & 0 & -2 \\ -4 & 4 & 0 & 8 \\ -2 & 1 & 2 & 4 \end{bmatrix}$ is

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