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 \(A^{13}\)?
Answer
(D)
Explanation
Squaring \(A\) gives a shortcut:
\[ A^2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = 2I, \qquad A^{13} = (A^2)^6 A = 2^6 A = 64A. \]
The eigenvalues of \(A\) satisfy \(\det(A - \lambda I) = (1 - \lambda)(-1 - \lambda) - 1 = \lambda^2 - 2 = 0\), so \(\lambda = \pm\sqrt{2}\). Hence \(A^{13} = 64A\) has eigenvalues \(\pm 64\sqrt{2}\), the same as \((\pm\sqrt{2})^{13} = \pm 2^6\sqrt{2}\).
Answer: D