How 2 prove complex conjugates using eigenvectors/matrices?

Suppose U is a matrix with real entries. Then the characteristic equation has real coefficients. Now suppose L is a complex eigenvalue with X the corresponding eigenvector, so UX = LX. Taking complex conjugates, we have UX* = L*X*. Thus, X* is a complex eigenvector corresponding to L*. Therefore, L* is an eigenvvalue. This proves: If p(x) = 0 is a polynomial equation with real coefficients and a is a complex root, then a* is also a root.