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 an orthogonal matrix , .
(A) . Correct.
(B) If is an eigenvalue, . Then . Since is orthogonal, , so . Correct.
(D) . Also . Correct.
(C) is incorrect because for all , including distinct ones.
(A) . Correct.
(B) If is an eigenvalue, . Then . Since is orthogonal, , so . Correct.
(D) . Also . Correct.
(C) is incorrect because for all , including distinct ones.
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