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GATE DA 2026 Q21: Power of a rotation matrix by 2π/5

Question

Let M=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)M = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} be a 2×22 \times 2 matrix, where θ=2π5\theta = \dfrac{2\pi}{5}, and I2=(1001)I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

Which of the following options is equal to M2026M^{2026}?

Try it: multiple choice: exactly one correct option

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Answer: B (official GATE 2026 final answer key)

Concept used: MkM^k is rotation by kθk\theta; rotating by a multiple of 2π2\pi gives the identity.

First step: M5M^5 rotates by 2π2\pi, so M5=I2M^5 = I_2; reduce 2026 mod 52026 \bmod 5.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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