An n × n matrix A with real entries satisfies the property: Ax ^2 = x ^2, for all x ∈ R^n, where · denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

A = I, where I denotes the identity matrix, is the only solution

The eigenvalues of A are either +1 or -1

GATE DA 2025 Q52: Matrices that preserve the Euclidean norm

Linear Algebra
2 marks
MSQ
No negative marking

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

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, D (official GATE 2025 final answer key)

Concept used: Norm-preserving real matrices are orthogonal: $A^{\top}A = I$ .

First step: Write $\lVert Ax \rVert^2 = x^{\top}A^{\top}Ax$ and compare with $x^{\top}x$ for all $x$ . For option C, consider a $2 \times 2$ rotation by $90^\circ$ .

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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 Q52: Matrices that preserve the Euclidean norm

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