Let n > 1. Consider an n × n matrix M with its elements from R. Let the vector (0, 1, 0, 0, , 0) ∈ R^n be in the null space of M. Which of the following options is/are always correct?

There are at least two non-zero vectors in the null space of M

GATE CSE 2026 Session 1 Q20: Matrix with e2 in its null space: determinant and rank

Engineering Mathematics
1 mark
MSQ
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Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \ldots, 0) \in \mathbb{R}^n$ be in the null space of $M$ .

Which of the following options is/are always correct?

Try it: multiple select: one or more correct options

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

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Answer: B, D (official GATE 2026 final answer key)

Concept used: A non-trivial null space means $M$ is singular; a null space is a subspace, closed under scaling.

First step: $M e_2 = 0$ with $e_2 \ne 0$ , so $M$ is singular; also $2e_2$ is in the null space.

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Source: official GATE 2026 Session 1 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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GATE CSE 2026 Session 1 Q20: Matrix with e2 in its null space: determinant and rank

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