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GATE DA 2025 Q37: If A³ = A, what is always true?

Question

Let A∈Rn×nA \in \mathbb{R}^{n \times n} be such that A3=AA^3 = A. Which one of the following statements is ALWAYS correct?

Try it: multiple choice: exactly one correct option

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Answer: D (official GATE 2025 final answer key)

Concept used: Rank of matrix products: rank⁡(AB)≤min⁡(rank⁡A,rank⁡B)\operatorname{rank}(AB) \le \min(\operatorname{rank} A, \operatorname{rank} B).

First step: Test the options with simple matrices that satisfy A3=AA^3 = A, such as A=IA = I and A=0A = 0. For the rank, note rank⁡(A)=rank⁡(A3)≤rank⁡(A2)≤rank⁡(A)\operatorname{rank}(A) = \operatorname{rank}(A^3) \le \operatorname{rank}(A^2) \le \operatorname{rank}(A).

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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