GATE DA 2025 | Question 37

MCQ · 2 marks · −0.66

Let \(A \in \mathbb{R}^{n \times n}\) be such that \(A^{3} = A\). Which one of the following statements is ALWAYS correct?

  1. Option A: \(A\) is invertible
  2. Option B: Determinant of \(A\) is \(0\)
  3. Option C: The sum of the diagonal elements of \(A\) is \(1\)
  4. Option D: \(A\) and \(A^{2}\) have the same rank (correct)

Answer

(D)

Explanation

The zero matrix and the identity both satisfy \(A^{3} = A\), and they rule out the first three options: \(A = 0\) is not invertible (A), \(A = I\) has determinant 1 (B), and \(A = 0\) has trace 0 (C).

For (D), multiplying by a matrix never increases rank, so \(\operatorname{rank}(A^{2}) \le \operatorname{rank}(A)\). Also \(A = A^{3} = A \cdot A^{2}\), so \(\operatorname{rank}(A) \le \operatorname{rank}(A^{2})\). The two ranks are therefore always equal.

Another view: \(A\) is diagonalisable with eigenvalues in \(\{-1, 0, 1\}\), and squaring keeps exactly the same eigenvalues non-zero.

Answer: D