Matrix $A$ with characteristic polynomial

Question

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 $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ ?

Answer 1

Hint: You know $$\rho(A-5I)=4-1=3$$ (why ?) and that $$\rho(A+2I)=4$$ (why ?)

What do you conclude from that, given the work you already did ?