Let A be a $5 \times 5$ matrix with 4 distinct eigenvalues. Two of its eigenspaces are unidimensional and one of its eigenspaces is bidimensional.
Does the characteristic polynomial of A has a root of multiplicity two?
I think it has. My reasoning is that because an eigenspace is bidimensional, one eigenvalue has geometric multiplicity of two. This implies that the algebraic multiplicity of this eigenvector is (at least) two. From this we may conclude that a root of the characteristic polynomial has multiplicity of at least two. As there are 5 roots and 4 distinct roots, the root which we know has multiplicity of at least two has multiplicity of exactly 2.