GATE DA 2024 Set 1 — Question 48

We analyze the solutions based on the rank and dimensions of matrix $M$.(A) For a square matrix $M \in \mathbb{R}^{3 \times 3}$, if $Mx=p$ has a unique solution, $M$ must be invertible (full rank). If $M$ is invertible, $Mx=q$ must also have a unique solution for any $q$. It cannot have infinite solutions. False.(B) For $M \in \mathbb{R}^{3 \times 3}$, if $Mx=p$ has no solution, $M$ is singular (rank < 3). If rank < 3, the nullity is at least 1. If we choose $q$ in the column space of $M$, the system is consistent and will have infinite solutions due to the non-zero nullity. True.(C) For $M \in \mathbb{R}^{2 \times 3}$ (2 equations, 3 variables), the maximum rank is 2. The number of free variables is $3 - \text{rank} \ge 1$. Thus, if a solution exists, it must be infinite. A unique solution is impossible. False.(D) For $M \in \mathbb{R}^{3 \times 2}$ (3 equations, 2 variables), the maximum rank is 2. If rank is 2, the columns are linearly independent, so if a solution exists (i.e., $p$ is in the column space), it is unique. Since the column space is a 2D plane in $\mathbb{R}^3$, there exist vectors $q$ outside this plane, leading to no solution. True.