GATE DA 2024 | Question 48

MSQ · 2 marks

Which of the following statements is/are TRUE?

Note: \(\mathbb{R}\) denotes the set of real numbers.

  1. Option A: There exist \(\boldsymbol{M} \in \mathbb{R}^{3 \times 3}\), \(\boldsymbol{p} \in \mathbb{R}^{3}\), and \(\boldsymbol{q} \in \mathbb{R}^{3}\) such that \(\boldsymbol{M}\mathbf{x} = \boldsymbol{p}\) has a unique solution and \(\boldsymbol{M}\mathbf{x} = \boldsymbol{q}\) has infinite solutions.
  2. Option B: There exist \(\boldsymbol{M} \in \mathbb{R}^{3 \times 3}\), \(\boldsymbol{p} \in \mathbb{R}^{3}\), and \(\boldsymbol{q} \in \mathbb{R}^{3}\) such that \(\boldsymbol{M}\mathbf{x} = \boldsymbol{p}\) has no solutions and \(\boldsymbol{M}\mathbf{x} = \boldsymbol{q}\) has infinite solutions. (correct)
  3. Option C: There exist \(\boldsymbol{M} \in \mathbb{R}^{2 \times 3}\), \(\boldsymbol{p} \in \mathbb{R}^{2}\), and \(\boldsymbol{q} \in \mathbb{R}^{2}\) such that \(\boldsymbol{M}\mathbf{x} = \boldsymbol{p}\) has a unique solution and \(\boldsymbol{M}\mathbf{x} = \boldsymbol{q}\) has infinite solutions.
  4. Option D: There exist \(\boldsymbol{M} \in \mathbb{R}^{3 \times 2}\), \(\boldsymbol{p} \in \mathbb{R}^{3}\), and \(\boldsymbol{q} \in \mathbb{R}^{3}\) such that \(\boldsymbol{M}\mathbf{x} = \boldsymbol{p}\) has a unique solution and \(\boldsymbol{M}\mathbf{x} = \boldsymbol{q}\) has no solutions. (correct)

Answer

(B), (D)

Explanation

A consistent system \(\boldsymbol{M}\mathbf{x} = \boldsymbol{b}\) has a unique solution exactly when the null space of \(\boldsymbol{M}\) is \(\{\mathbf{0}\}\); otherwise every consistent right-hand side has infinitely many solutions.

  • (A) False. A unique solution forces a trivial null space, so the square \(\boldsymbol{M}\) is invertible and \(\boldsymbol{M}\mathbf{x} = \boldsymbol{q}\) also has exactly one solution.
  • (B) True. Take \(\boldsymbol{M} = \operatorname{diag}(1, 1, 0)\), \(\boldsymbol{p} = (0, 0, 1)^{\top}\) (no solution) and \(\boldsymbol{q} = \mathbf{0}\) (infinitely many).
  • (C) False. With 3 unknowns and rank at most 2, the null space is never trivial, so no solution is unique.
  • (D) True. Let \(\boldsymbol{M}\) have columns \((1, 0, 0)^{\top}\) and \((0, 1, 0)^{\top}\). Then \(\boldsymbol{p} = (1, 1, 0)^{\top}\) has the single solution \((1, 1)\), while \(\boldsymbol{q} = (0, 0, 1)^{\top}\) lies outside the column space.

Answer: B, D