I'm trying to solve this issue:
Given A, a 4x4 singular Matrix. It is known that $\rho(A+2I)=2$ and $|A-2I| =0$.
Find the characteristic polynomial of A, is A similar to a diagonal matrix?
I've found that because A is singular, 0 is an eigenvalue to A.
And because $|A-2I| = 0$, $|2I-A| = 0$ and 2 is also an eigenvalue.
I'm not sure what to do with the information of the rank. It would really help if I could find that $\rho(-2I-A) = 2$, but I don't know if that something I can claim.
Given that information, the geometric multiplicity of -2 is 2, and we have the eigen values of 0 and 2, which would mean the matrix is similar to a diagonal matrix. And the polynomial would be $p(x) = x(x-2)(x+2)^2$ Am I on the right path? Thanks!
EDIT: Fixed "algebraic" to "geometric" in the last paragraph.