GATE EE 2026 Set 1 — Question 55
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Engineering Mathematics → Linear Algebra → Determinants
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Question
Consider an orthogonal matrix with real entries and each column having unit Euclidean norm.Which of the following statements is/are correct?
Correct answer
(A) The value of the determinant of A is either +1 or -1; (B) The eigenvalues of A have modulus 1; (D) \ Ax \ = \ x \, for all x ∈ Rⁿ, where \ x \ denotes the Euclidean norm of x, and (Ax)^T(Ay) = x^Ty, for all x, y ∈ Rⁿ
Solution
For a real orthogonal matrix , the columns are orthonormal, so (equivalently ).(A) . Correct.(B) An eigenvalue of a real matrix may be complex, with a complex eigenvector : . Because is real, , so andBut , so . Correct.(D) For real :and with this gives . Both parts hold for all . Correct.(C) Its first part is true, but its second part claims for distinct , which contradicts the identity just shown. Incorrect.Answer: (A), (B), (D)
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