GATE 2025 DA Q.38: solution and answer

GATE 2025 · DAQ.38 · 2 marks · MCQ

Let ${x_{1}, x_{2}, \dots, x_{n}}$ be a set of linearly independent vectors in $R^{n}$ . Let the $(i, j)$ -th element of matrix $A \in R^{n \times n}$ be given by $A_{ij} = x_{i}^{⊤} x_{j}$ , $1 \leq i, j \leq n$ . Which one of the following statements is correct?

(A)
$A$ is invertible
(B)
$0$ is a singular value of $A$
(C)
Determinant of $A$ is $0$
(D)
$z^{⊤} A z = 0$ for some non-zero $z \in R^{n}$

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Answer and solution

ANSWER

The correct option is (A): $A$ is invertible.

Official answer: ASource: Official answer key, GATE 2025 DA (IIT Roorkee)

Let $X = [x_{1} x_{2} \dots x_{n}] \in R^{n \times n}$ have the given vectors as its columns. The $(i, j)$ entry of $X^{⊤} X$ is $x_{i}^{⊤} x_{j}$ , which is $A_{ij}$ . So $A = X^{⊤} X$ (the Gram matrix of the vectors).
The $n$ vectors are linearly independent in $R^{n}$ , so $X$ is square with full column rank. Hence $rank (X) = n$ and $det X \neq = 0$ .
Then $det A = det (X^{⊤}) det (X) = (det X)^{2} > 0$ . So $A$ is invertible and $det A \neq = 0$ . This makes (A) true and (C) false.
The singular values of $A$ are the square roots of the eigenvalues of $A^{⊤} A$ . Because $A$ is invertible, $A^{⊤} A$ is invertible too, so none of its eigenvalues is $0$ . Hence $0$ is not a singular value of $A$ , and (B) is false.
For any non-zero $z \in R^{n}$ , $z^{⊤} A z = z^{⊤} X^{⊤} X z = ∥ X z ∥^{2}$ . Since $X$ is invertible, $X z \neq = 0$ , so $z^{⊤} A z > 0$ . Thus $A$ is positive definite and $z^{⊤} A z = 0$ happens only for $z = 0$ . So (D) is false.
Only (A) is correct.

KEY IDEA

The matrix with entries $x_{i}^{⊤} x_{j}$ is the Gram matrix $A = X^{⊤} X$ . When the columns of $X$ are linearly independent, $X$ is invertible, so $A$ is symmetric positive definite and in particular invertible.

WATCH OUT FOR

Treating "Gram matrix" as automatically singular, or mixing up positive semidefinite with positive definite. A Gram matrix is only semidefinite in general. It is singular exactly when the vectors are linearly dependent, and here they are independent.

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GATE 2025 DA Q.38: solution and answer

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