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GATE 2025 DA – Question 52

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

An $n \times n$ matrix $A$ with real entries satisfies the property: $\|Ax\|_2 = \|x\|_2$, for all $x \in \mathbb{R}^n$, where $\|\cdot\|$ denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

  1. $A$ must be orthogonal
  2. $A = I$, where $I$ denotes the identity matrix, is the only solution
  3. The eigenvalues of $A$ are either $+1$ or $-1$
  4. $A$ has full rank

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Show answer and explanation

Correct answer: (A) $A$ must be orthogonal; (D) $A$ has full rank

Explanation

$\|Ax\| = \|x\|$ for all $x$ means $x^\top A^\top Ax = x^\top x$ for all $x$, so $A^\top A = I$ and $A$ is orthogonal (A is true). Such a matrix is invertible, with full rank (D is true). Rotations and reflections also satisfy the property, so $I$ is not the only solution (B is false), and a rotation by 90° has the complex eigenvalues $\pm i$, so the eigenvalues need not be $\pm 1$ (C is false).