GATE CSE 2024 Session 1 Q49: Solutions of Ax = 0 for a wide n × m matrix

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 $Ax = 0$ ?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A (official GATE 2024 final answer key)

Concept used: Rank–nullity: the null space has dimension $m - \text{rank}(A) \ge m - n$ .

First step: Note the rank is at most $n$ but may be smaller; test (B) on the zero matrix and (C), (D) on small examples such as $A = [1 ;; 1]$ .

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Source: official GATE 2024 Session 1 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key