An n x n matrix A with real entries satisfies the property: ||Ax||^2 = ||x||^2, for all z in R^n, where ||.|| denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct? A. A^T * A = I B. Determinant of A is 0 C. The sum of the diagonal elements of A is n D. A * A^T = I Select all that apply.

GATE 2025 · Engineering Mathematics · Matrix · medium

Answer: Options A and D are always correct. Answer: A, D

1. Derive A^T A = I from the norm condition: ||Ax||^2 = ||x||^2 means x^T(A^T A)x = x^T x for all x in R^n. This holds for all x if and only if A^T A = I. So option A is ALWAYS correct.
2. Derive A A^T = I for square A: Since A is n x n (square) and A^T A = I, A^T is a left inverse of A. For finite-dimensional square matrices, left inverse = right inverse. So A A^T = I. Option D is ALWAYS correct.
3. Check option B: det(A) = 0?: det(A) = +1 or -1, never 0. Option B is WRONG.
4. Check option C: trace(A) = n?: For A = -I_n (which is orthogonal since (-I)^T(-I) = I^2 = I), trace = -n != n. Option C is WRONG.