The GATE Grind

GATE 2025 DA – Question 37

Linear Algebra · Vector spaces, linear independence, rank and nullity · 2 marks · Multiple choice

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

  1. $A$ is invertible
  2. Determinant of $A$ is 0
  3. The sum of the diagonal elements of $A$ is 1
  4. $A$ and $A^2$ have the same rank

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Correct answer: (D) $A$ and $A^2$ have the same rank

Explanation

The zero matrix and the identity matrix both satisfy $A^3 = A$, which rules out A (the zero matrix is not invertible), B (the identity has determinant 1) and C (the trace of the zero matrix is 0). For D, $A = A^3 = A^2 \cdot A$ gives $\text{rank}(A) \le \text{rank}(A^2)$, and always $\text{rank}(A^2) \le \text{rank}(A)$. So the two ranks are equal.