Question-99

Let \(\displaystyle A=\begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}\) be a \(\displaystyle 2\times 2\) matrix. What are the eigenvalues of \(\displaystyle A^{13}\)?

• \(\displaystyle 1,-1\)
• \(\displaystyle 2\sqrt{2} ,-2\sqrt{2}\)
• \(\displaystyle 4\sqrt{2} ,-4\sqrt{2}\)
• \(\displaystyle 64\sqrt{2} ,-64\sqrt{2}\)

NoteAnswer

• \(\displaystyle 1,-1\)
• \(\displaystyle 2\sqrt{2} ,-2\sqrt{2}\)
• \(\displaystyle 4\sqrt{2} ,-4\sqrt{2}\)
• \(\displaystyle 64\sqrt{2} ,-64\sqrt{2}\)

NoteSolution

If \(\displaystyle \lambda\) is an eigenvalue of \(\displaystyle A\), then \(\displaystyle \lambda ^{13}\) is an eigenvalue of \(\displaystyle A^{13}\). To get the eigenvalues of \(\displaystyle A\):

\[ \begin{aligned} -( 1-\lambda )( 1+\lambda ) -1 & =0\\ \lambda ^{2} -2 & =0\\ \lambda & =\pm \sqrt{2} \end{aligned} \]

Therefore, the eigenvalues of \(\displaystyle A^{13}\) are \(\displaystyle 64\sqrt{2} ,-64\sqrt{2}\).