Write \(A = P C P^{-1}\) for the matrix \[A = \begin{bmatrix} 0 & 1 \\ -10 & 6 \end{bmatrix},\] with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).
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Write \(A = P C P^{-1}\) for the matrix \[A = \begin{bmatrix} 0 & 1 \\ -10 & 6 \end{bmatrix},\] with \(C\) a rotation matrix corresponding to a complex eigenvalue \(\lambda = a + b i\).
\begin{solution} Solution: \[\lambda = 3 + 1i\] \[C = \begin{bmatrix} 3 & 1 \\ -1 & 3 \end{bmatrix}\] \[A = \begin{bmatrix} -1 & 0 \\ -3 & -1 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ -1 & 3 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 3 & -1 \end{bmatrix}\] \end{solution}