Can eigenvectors of a real matrix with complex eigenvalues be linearly independent?

Question

I am a bit confused about whether a real matrix with complex eigenvalues can have linearly independent eigenvectors? I do not understand how we can invert this real matrix if it has complex conjugates eigenvalues.

Answer 1

A matrix some of whose eigenvalues are complex will not generally have a basis of real eigenvectors (example: the rotation matrix $\begin{pmatrix} \cos \theta & \sin \theta\\ -\sin \theta & \cos \theta\end{pmatrix}.$ As for inverses, these have nothing to do with eigenvalues, so use Cramer's rule, and be happy.