Chapter 5.5, Problem 1PP

Show that if a and b are real, then the eigenvalues of A = $\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$ are a ± bi, with corresponding eigenvectors $\left[\begin{array}{l}1\\ -i\end{array}\right]$ and $\left[\begin{array}{l}1\\ i\end{array}\right]$.