GATE CS 2025 Set 1 Q.41: Linear Algebra

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 $A^{13}$ ?

$1, -1$

$2\sqrt{2}, -2\sqrt{2}$

$4\sqrt{2}, -4\sqrt{2}$

$64\sqrt{2}, -64\sqrt{2}$

Official answer and solution

Official answer: D
From: Official GATE 2025 answer key

Solution

Find the eigenvalues $\lambda_1, \lambda_2$ of $A$ (or note that $A^2 = 2I$ ) and raise them to the 13th power.

The characteristic polynomial of $A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$ is:

\det(A - \lambda I) = (1 - \lambda)(-1 - \lambda) - 1 = \lambda^2 - 2 = 0 \implies \lambda = \sqrt{2}, -\sqrt{2}

If $\lambda$ is an eigenvalue of $A$ , then $\lambda^{13}$ is an eigenvalue of $A^{13}$ . Since $(\sqrt{2})^{13} = (\sqrt{2})^{12} \sqrt{2} = 2^6 \sqrt{2} = 64\sqrt{2}$ , the eigenvalues of $A^{13}$ are:

64\sqrt{2}, \; -64\sqrt{2}

Final answer

Option (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 41 of 65
Section: Computer Science and Information Technology

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