Let n > 1. Consider an n n matrix M with its elements from R. Let the vector (0, 1, 0, 0, , 0) Rn be in the null space of M. Which of the following options is/are always correct? A) Determinant of M is 1 B) Determinant of M is 0 C) Rank of M is 1 D) There are at least two non-zero vectors in the null space of M

Official answer: B) Determinant of M is 0; D) There are at least two non-zero vectors in the null space of M. Explanation: Let v = (0, 1, 0, , 0)T Rn. 1. Determinant of M: Since v is a non-zero vector and v Null(M), we have Mv = 0. This implies that the null space of M has dimension at least 1 (i.e., Nullity(M) 1). Therefore, 0 is an eigenvalue of M, which means: (M) = 0 Hence, (B) is correct and (A) is incorrect. 2. Rank of M: By the Rank-Nullity Theorem, Rank(M) = n - Nullity(M) n - 1. For an arbitrary n > 1, Rank(M) is not necessarily 1. Hence, (C) is incorrect. 3. Vectors in the Null Space: Since Null(M) is a subspace over R, for any scalar c R, c v Null(M). Since R contains infinitely many non-zero scalars, there exist infinitely many distinct non-zero vectors in Null(M) (for example, v and 2v). Thus, there are at least two non-zero vectors in the null space of M. Hence, (D) is correct. Therefore, the correct options are B and D.

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

Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Roorkee · Audited Aug 2026

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GATE CSE 2026 Set 1 Q20 · MSQ · 1 mark

Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$ . Which of the following options is/are always correct?

Select one or more options

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Linear Algebra

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How GATE tests Linear Algebra

Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and eigenvectors). Questions heavily leverage algebraic identities such as trace-sum, determinant-product relations, rank-nullity theorem, and structural properties like orthogonality of symmetric matrices and LU decomposition. Problems frequently appear as 1-mark and 2-mark conceptual MCQs/MSQs or direct numerical computation NATs.

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Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Roorkee · Audited Aug 2026

Take topic test

GATE CSE 2026 Set 1 Q20 · MSQ · 1 mark

Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$ . Which of the following options is/are always correct?

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$ . Which of the following options is/are always correct?

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Let n>1n > 1n>1. Consider an n×nn \times nn×n matrix MMM with its elements from R\mathbb{R}R. Let the vector (0,1,0,0,…,0)∈Rn(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n(0,1,0,0,…,0)∈Rn be in the null space of MMM. Which of the following options is/are always correct?

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Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$ . Which of the following options is/are always correct?

Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$ . Which of the following options is/are always correct?