GATE 2025 DA Q.52: solution and answer

GATE 2025 · DAQ.52 · 2 marks · MSQ

An $n \times n$ matrix $A$ with real entries satisfies the property: $∥ A x ∥^{2} = ∥ x ∥^{2}$ , for all $x \in R^{n}$ , where $∥ \cdot ∥$ denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

(A)
$A$ must be orthogonal
(B)
$A = I$ , where $I$ denotes the identity matrix, is the only solution
(C)
The eigenvalues of $A$ are either $+ 1$ or $- 1$
(D)
$A$ has full rank

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Answer and solution

ANSWER

The statements that are always correct are (A) and (D).

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

Expand the condition: $∥ A x ∥^{2} = (A x)^{T} (A x) = x^{T} A^{T} A x$ , so the given property says $x^{T} (A^{T} A - I) x = 0$ for every $x \in R^{n}$ .
Let $M = A^{T} A - I$ . It is symmetric, since $(A^{T} A)^{T} = A^{T} A$ . Take $x = e_{i}$ (a standard basis vector): $e_{i}^{T} M e_{i} = M_{ii} = 0$ . Take $x = e_{i} + e_{j}$ : $0 = (e_{i} + e_{j})^{T} M (e_{i} + e_{j}) = M_{ii} + M_{j j} + 2 M_{ij} = 2 M_{ij}$ , so $M_{ij} = 0$ . Hence $M = 0$ , that is $A^{T} A = I$ .
$A^{T} A = I$ is the definition of an orthogonal matrix, so (A) is always correct.
(B) is false. Every orthogonal matrix satisfies the property, not only $I$ . For example $A = [01 - 1 0]$ has $A^{T} A = I$ but $A \neq = I$ (a rotation by $9 0^{\circ}$ ); so does the reflection $[10 0 - 1]$ .
(C) is false. The rotation above has characteristic equation $λ^{2} + 1 = 0$ , so its eigenvalues are $\pm i$ , which are neither $+ 1$ nor $- 1$ . In general a real orthogonal matrix has eigenvalues of modulus $1$ , but they need not be real.
(D) is true. $det (A)^{2} = det (A^{T}) det (A) = det (A^{T} A) = det (I) = 1$ , so $det A = \pm 1 \neq = 0$ . Thus $A$ is invertible and $rank (A) = n$ , i.e. full rank.
Correct options: (A) and (D).

KEY IDEA

$∥ A x ∥^{2} = ∥ x ∥^{2}$ for all $x$ forces $A^{T} A = I$ (a symmetric matrix whose quadratic form vanishes for every $x$ is zero). So $A$ is orthogonal, hence invertible with $det A = \pm 1$ . Eigenvalues of an orthogonal matrix have modulus 1 but can be complex.

WATCH OUT FOR

Choosing (C) because orthogonal matrices have eigenvalues of absolute value 1. Over the reals they can be complex, as for a rotation with eigenvalues $\pm i$ . Another slip is choosing (B) by reading the property as 'A leaves every vector unchanged'. It only says that lengths are preserved.

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GATE 2025 DA Q.52: solution and answer

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