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GATE 2024 CS (CS1) – Question 49

Engineering Mathematics · Linear Algebra · 2 marks · Multiple select

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. There exist at least $m-n$ linearly independent solutions to this system
  2. There exist $m-n$ linearly independent vectors such that every solution is a linear combination of these vectors
  3. There exists a non-zero solution in which at least $m-n$ variables are 0
  4. There exists a solution in which at least $n$ variables are non-zero

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Show answer and explanation

Correct answer: (A) There exist at least $m-n$ linearly independent solutions to this system; (C) There exists a non-zero solution in which at least $m-n$ variables are 0

Explanation

Nullity ≥ m−n, so at least m−n independent solutions exist (A). A basic solution sets at least m−n free variables... yielding a non-zero solution with at least m−n zeros (C). B fails since nullity can exceed m−n; D is not guaranteed (e.g., A=0 style cases).