Let \x_1, x_2, , x_n\ be a set of linearly independent vectors in R^n. Let the (i, j)-th element of matrix A ∈ R^n × n be given by A_ij = x_i^x_j, 1 ≤ i, j ≤ n. Which one of the following statements is correct?

z^Az = 0 for some non-zero z ∈ R^n

GATE DA 2025 Q38: Gram matrix of linearly independent vectors

Linear Algebra
2 marks
MCQ
−2/3 if wrong

Let $\{x_1, x_2, \ldots, x_n\}$ be a set of linearly independent vectors in $\mathbb{R}^n$ . Let the $(i, j)$ -th element of matrix $A \in \mathbb{R}^{n \times n}$ be given by $A_{ij} = x_i^{\top}x_j$ , $1 \le i, j \le n$ . Which one of the following statements is correct?

Try it: multiple choice: exactly one correct option

Wrong answer costs 2/3 mark in GATE.

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Answer: A (official GATE 2025 final answer key)

Concept used: A Gram matrix $A = X^{\top}X$ of linearly independent columns is positive definite.

First step: Put the vectors as columns of $X$ , so $A = X^{\top}X$ . Then $z^{\top}Az = \lVert Xz \rVert^2$ , which is zero only if $Xz = 0$ .

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE DA 2025 Q38: Gram matrix of linearly independent vectors

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