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(B) The given matrix is $A = \begin{bmatrix} 1 & 1 & 1 & 3 \ 2 & 2 & -1 & 3 \ 1 & 1 & -1 & 1 \end{bmatrix}$.Applying row operations to reduce the ma

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$2$

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(B) The given matrix is $A = \begin{bmatrix} 1 & 1 & 1 & 3 \ 2 & 2 & -1 & 3 \ 1 & 1 & -1 & 1 \end{bmatrix}$.Applying row operations to reduce the matrix to row-echelon form:Perform $R_2 ightarrow R_2 - 2R_1$ and $R_3 ightarrow R_3 - R_1$:$A \sim \begin{bmatrix} 1 & 1 & 1 & 3 \ 0 & 0 & -3 & -3 \ 0 & 0 & -2 & -2 \end{bmatrix}$.Perform $R_2 ightarrow -\frac{1}{3}R_2$ and $R_3 ightarrow -\frac{1}{2}R_3$:$A \sim \begin{bmatrix} 1 & 1 & 1 & 3 \ 0 & 0 & 1 & 1 \ 0 & 0 & 1 & 1 \end{bmatrix}$.Perform $R_3 ightarrow R_3 - R_2$:$A \sim \begin{bmatrix} 1 & 1 & 1 & 3 \ 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 0 \end{bmatrix}$.The number of non-zero rows in the row-echelon form is $2$.Therefore,the rank of the matrix $A$ is $2$.

The rank of the matrix $A=\begin{bmatrix} 1 & 1 & 1 & 3 \\ 2 & 2 & -1 & 3 \\ 1 & 1 & -1 & 1 \end{bmatrix}$ is

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