QS-023 · Linear Algebra · Easy

NPTEL GATE DA MOCK | Week 1 - 61

Let $A$ be the $2\times2$ matrix $$ A= \begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}. $$ What are the eigenvalues of $A^{13}$? A. $64\sqrt{2},-64\sqrt{2}$ B. $64,-64$ C. $128, -128$ D. $2^{13},-2^{13}$

Solution The characteristic equation of $A$ is $$ \det(A-\lambda I)= \begin{vmatrix} 1-\lambda & 1\\ 1 & -1-\lambda \end{vmatrix}=0. $$ Therefore, $$ (1-\lambda)(-1-\lambda)-1=0 $$ $$ \lambda^2-2=0. $$ Hence, the eigenvalues of $A$ are $$ \lambda_1=\sqrt{2},\qquad \lambda_2=-\sqrt{2}. $$ If $\lambda$ is an eigenvalue of $A$, then $\lambda^{13}$ is an eigenvalue of $A^{13}$. Thus, $$ (\sqrt{2})^{13}=2^6\sqrt{2}, $$ and $$ (-\sqrt{2})^{13}=-2^6\sqrt{2}. $$ Hence, the eigenvalues of $A^{13}$ are $$ \boxed{64\sqrt{2},-64\sqrt{2}}. $$

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