$$\text{Let } M \text{ be a } 5 \times 5 \text{ matrix with real entries such that } \\ \text{Rank}(M) = 3. \\ \text{Consider the linear system } Mx = b. \\ \text{Let the row-reduced echelon form of the augmented matrix } [M \, b] \\ \text{be } R, \text{ and let } R[i, :] \text{ denote the } i\text{-th row of } R. \\ \text{Suppose that the linear system admits a solution. Which of the following statements is necessarily true?}$$​

Correct option is D

​$$\text{Let } M = [a_{ij}]_{5 \times 5} \text{ with all } a_{ij} \in \mathbb{R}. \\[10pt]\text{If } \text{Rank}(M) = 3, \text{ then the row-reduced echelon form of } M \text{ will look like:} \\[10pt]\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{bmatrix} \\[10pt]\text{Now, if } Mx = b \text{ is the given system and admits a solution:} \\[10pt]\text{As the system admits a solution, } \text{Rank}(M) = \text{Rank}([M \, b]). \\[10pt]\text{The augmented matrix } [M \, b] \text{ will be given by:} \\[10pt]\begin{bmatrix}1 & 0 & 0 & 0 & 0 & b_1 \\0 & 1 & 0 & 0 & 0 & b_2 \\0 & 0 & 1 & 0 & 0 & b_3 \\0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0\end{bmatrix} \\[10pt]\text{Thus, } R[4, :] = [0 \, 0 \, 0 \, 0 \, 0 \, 0].$$​​