GATE DA 2025 Set 1 — Question 52
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Linear Algebra → Eigenvalues & Eigenvectors → Eigenvalue Computation
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Question
An matrix with real entries satisfies the property: , for all , where denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?
Correct answer
(A) A must be orthogonal; (D) A has full rank
Solution
Given for all .
Squaring both sides:Since this holds for all , and is symmetric, we must have , which implies . Thus, is an orthogonal matrix.(A) True: is orthogonal.
(B) False: Any orthogonal matrix satisfies the condition, not just (e.g., a rotation matrix).
(C) False: The eigenvalues of an orthogonal matrix have modulus 1, but they can be complex (e.g., ). They are not restricted to unless is symmetric.
(D) True: Since , . Thus, is invertible and has full rank.
Squaring both sides:Since this holds for all , and is symmetric, we must have , which implies . Thus, is an orthogonal matrix.(A) True: is orthogonal.
(B) False: Any orthogonal matrix satisfies the condition, not just (e.g., a rotation matrix).
(C) False: The eigenvalues of an orthogonal matrix have modulus 1, but they can be complex (e.g., ). They are not restricted to unless is symmetric.
(D) True: Since , . Thus, is invertible and has full rank.
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