The GATE Grind

GATE 2026 DA – Question 21

Linear Algebra · Eigenvalues, eigenvectors and determinant · 1 mark · Multiple choice

Let $M = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$ be a $2 \times 2$ matrix, where $\theta = \frac{2\pi}{5}$, and $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

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

  1. $M^2$
  2. $M$
  3. $M^{-1}$
  4. $I_2$

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Show answer and explanation

Correct answer: (B) $M$

Explanation

$M$ is a rotation by the angle $\theta = \frac{2\pi}{5}$ and $M^k$ is a rotation by $k\theta$. So $M^5$ is a rotation by $2\pi$, which is $I_2$. Since $2026 = 5 \times 405 + 1$, $M^{2026} = (M^5)^{405}M = M$.