GATE CS 2024 Set 1 | Question 49

MSQ · 2 marks

Let \(A\) be any \(n \times m\) matrix, where \(m > n\). Which of the following statements is/are TRUE about the system of linear equations \(Ax = \mathbf{0}\) ?

  1. Option A: There exist at least \(m - n\) linearly independent solutions to this system (correct)
  2. Option B: There exist \(m - n\) linearly independent vectors such that every solution is a linear combination of these vectors
  3. Option C: There exists a non-zero solution in which at least \(m - n\) variables are 0
  4. Option D: There exists a solution in which at least \(n\) variables are non-zero

Answer

(A)

Explanation

Since \(A\) has only \(n\) rows, \(\operatorname{rank}(A) \le n\). By rank–nullity, the solutions of \(Ax = \mathbf{0}\) form a subspace of dimension \(m - \operatorname{rank}(A) \ge m - n \ge 1\).

  • (A) True: a basis of this subspace contains at least \(m - n\) linearly independent solutions.
  • (B) False: it needs the dimension to be exactly \(m - n\). For the \(1 \times 3\) zero matrix every vector in \(\mathbb{R}^{3}\) is a solution, and \(m - n = 2\) vectors cannot span a 3-dimensional space.
  • (C) False: for \(A = \begin{pmatrix} 1 & 1 \end{pmatrix}\) the solutions are \((t, -t)\), so no non-zero solution has a zero entry.
  • (D) False: for \(A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\) every solution is \((0, 0, t)\), with at most one non-zero entry rather than \(n = 2\).

Answer: A