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GATE 2025 DA – Question 38

Linear Algebra · Special matrices: projection, orthogonal and idempotent, quadratic forms · 2 marks · Multiple choice

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?

  1. $A$ is invertible
  2. 0 is a singular value of $A$
  3. Determinant of $A$ is 0
  4. $z^\top Az = 0$ for some non-zero $z \in \mathbb{R}^n$

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Correct answer: (A) $A$ is invertible

Explanation

$A = X^\top X$ where $X$ has the vectors $x_i$ as its columns. They are linearly independent, so $X$ is invertible. Then $z^\top Az = \|Xz\|^2 > 0$ for every non-zero $z$, so $A$ is positive definite. It is therefore invertible, with a non-zero determinant, and none of its singular values is 0. Only statement A is correct.