Let A be a 2 × 2 matrix as given. A = bmatrix 1 & 1 \\ 1 & -1 bmatrix What are the eigenvalues of the matrix A^13?

GATE CSE 2025 Session 1 Q41: Eigenvalues of A^13 for a 2×2 matrix

Let $A$ be a $2 \times 2$ matrix as given.

$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

What are the eigenvalues of the matrix $A^{13}$ ?

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Answer: D (official GATE 2025 final answer key)

Concept used: If $\lambda$ is an eigenvalue of $A$ , then $\lambda^k$ is an eigenvalue of $A^k$ .

First step: The characteristic equation is $\lambda^2 - 2 = 0$ , so $\lambda = \pm\sqrt{2}$ ; raise these to the 13th power.

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Source: official GATE 2025 Session 1 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key