Let A in R^(n x n) be such that A^4 = I. Which one of the following statements is ALWAYS correct? A. A is invertible. B. Determinant of A is 1. C. The sum of the diagonal elements of A is n. D. A and A^T have the same rank.

GATE 2025 · Engineering Mathematics · Matrix · medium

Answer: Option D: A and A^T always have the same rank. Answer: D

1. Check option A: Is A always invertible?: Yes, A^4 = I implies A is invertible. But the question might have nuance about option A vs D.
2. Check option B: Is det(A) always 1?: det(A) can be +1 or -1 (for real matrices). Example: A = -I has det(-I) = (-1)^n. So det(A) is NOT always 1. Option B is wrong.
3. Check option C: Is trace(A) always n?: If A = -I (n x n), trace = -n != n. So option C is wrong.
4. Check option D: Does rank(A) = rank(A^T) always?: This is a fundamental theorem of linear algebra: the row rank equals the column rank, and row rank of A = column rank of A = column rank of A^T = row rank of A^T. So rank(A) = rank(A^T) always holds for any matrix, including those satisfying A^4 = I.