Let A be a 2 × 2 matrix as given. A = 1 & 1 1 & -1 What are the eigenvalues of the matrix A13 ? A) 1, -1 B) 2√(2), -2√(2) C) 4√(2), -4√(2) D) 64√(2), -64√(2)

Official answer: D) 64√(2), -64√(2). Explanation: To find the eigenvalues of the matrix A, we solve the characteristic equation det(A - I) = 0: det 1 - & 1 1 & -1 - = 0 (1 - )(-1 - ) - (1)(1) = 0 2 - 1 - 1 = 0 2 - 2 = 0 = ±√(2) If is an eigenvalue of a matrix A, then n is an eigenvalue of An. For the matrix A13, the eigenvalues are: 113 = (√(2))13 = (√(2))12 · √(2) = (26)√(2) = 64√(2) 213 = (-√(2))13 = -(√(2))13 = -64√(2) Therefore, the eigenvalues of A13 are 64√(2), -64√(2), which corresponds to option (D).

Linear Algebra

GATE CSE 2025 Set 1 · Q41

MCQ2 marksEasy

Time00:00

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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Official answer: D) 64√(2), -64√(2)

Linear Algebra

To find the eigenvalues of the matrix $A$ , we solve the characteristic equation $\det(A - \lambda I) = 0$ :

\det\begin{bmatrix} 1 - \lambda & 1 \\ 1 & -1 - \lambda \end{bmatrix} = 0

(1 - \lambda)(-1 - \lambda) - (1)(1) = 0

\lambda^2 - 1 - 1 = 0 \implies \lambda^2 - 2 = 0 \implies \lambda = \pm\sqrt{2}

If $\lambda$ is an eigenvalue of a matrix $A$ , then $\lambda^n$ is an eigenvalue of $A^n$ .

For the matrix $A^{13}$ , the eigenvalues are:

\lambda_1^{13} = (\sqrt{2})^{13} = (\sqrt{2})^{12} \cdot \sqrt{2} = (2^6)\sqrt{2} = 64\sqrt{2}

\lambda_2^{13} = (-\sqrt{2})^{13} = -(\sqrt{2})^{13} = -64\sqrt{2}

Therefore, the eigenvalues of $A^{13}$ are $64\sqrt{2}, -64\sqrt{2}$ , which corresponds to option (D).

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This Linear Algebra previous year question is Q41 of the GATE CSE 2025 Set 1 paper. Find more GATE CSE PYQs on Linear Algebra in Engineering Mathematics, or browse all GATE CSE PYQs.

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