Let c1, , cn be scalars, not all zero, such that i=1n ci ai = 0 where ai are column vectors in Rn. Consider the set of linear equations Ax = b where A = [a1, , an] and b = i=1n ai. The set of equations has A) a unique solution at x = Jn where Jn denotes a n-dimensional vector of all 1 B) no solution C) infinitely many solutions D) finitely many solutions

Official answer: C) infinitely many solutions. Explanation: 1. Since there exist scalars c1, , cn, not all zero, such that i=1n ci ai = 0, the column vectors a1, , an of the n n matrix A are linearly dependent. 2. Linear dependence implies that rank(A) < n, so the homogeneous equation Ax = 0 has non-trivial solutions (the nullity of A is at least 1, with vector c = [c1, , cn]T 0 satisfying Ac = 0). 3. The vector b is given by: b = i=1n ai = A Jn where Jn = [1, 1, , 1]T. This shows that x = Jn is a particular solution to Ax = b, meaning the system is consistent. 4. Since the system Ax = b is consistent and rank(A) < n, it must have infinitely many solutions of the form x = Jn + k c for any scalar k R. Hence, the correct option is (C).

A) a unique solution at x = Jn where Jn denotes a n-dimensional vector of all 1

Engineering Mathematics · Linear Algebra

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GATE CSE 2017 Set 1 Q3 · MCQ · 1 mark

Let $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$ where $a_i$ are column vectors in $\mathbf{R}^n$ . Consider the set of linear equations $Ax = b$ where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^n a_i$ . The set of equations has

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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

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GATE CSE 2017 Set 1 Q3 · MCQ · 1 mark

Let $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$ where $a_i$ are column vectors in $\mathbf{R}^n$ . Consider the set of linear equations $Ax = b$ where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^n a_i$ . The set of equations has

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Key concept

Linear Algebra

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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 $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$ where $a_i$ are column vectors in $\mathbf{R}^n$ . Consider the set of linear equations $Ax = b$ where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^n a_i$ . The set of equations has

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Let $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$ where $a_i$ are column vectors in $\mathbf{R}^n$ . Consider the set of linear equations $Ax = b$ where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^n a_i$ . The set of equations has

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