Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$

Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix)

Prove: $A$ is diagonalizable.

I tried by saying first the the eigenvalues are $-3,1,5.$

Then, I know that their algebraic multiplicity of $-3$ is $2$, of $1$ is $1$, and of $5$ is $1$.

Now I need only to prove that the geometric multiplicity of $-3$ is $2$ to show that $A$ is diagonalizable.

How can I prove it by using $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ ?

• You know the rank of $A+2I$ and of $A-5I$. From that, compute that the rank of $A+3I$ is $2$. – Daniel Fischer Jun 14 '14 at 14:28
• I know that A's rank has to be 4 because the sum of the algebric multiplexing is 4. and also that A is singular. – Ilan Aizelman WS Jun 14 '14 at 14:33

Hint: You know $$\rho(A-5I)=4-1=3$$ (why ?) and that $$\rho(A+2I)=4$$ (why ?)
• Is it true that $\rho(A-5I)$ = 1 ? If yes, is it because its algebric multiplexing is 1? – Ilan Aizelman WS Jun 14 '14 at 14:35
• @IlanAizelmanWS - Yes. This would imply that the geometric multiplexing is $\leq 1$. But you also know that it is at least $1$ since it is an eigenvalue and so the geometric multiplexing is $1$ – Belgi Jun 14 '14 at 14:36