General question about eigenvalue, eigenvectors.

Question

I have the following question :

$A$ is a $n \times n$ matrix, and this is the characteristic polynom $$p(x)=(x+3)^2(x-1)(x-5)$$

Then I can conclude that $n=4$ since the number of the roots is $4$, now this is my question :

Can I conclude for any $\lambda \neq -3,1,5$ that $\lambda I-A$ and $A- \lambda I$ are invertible matrixs?

I think this statement is true, but I'd like to be sure of that, since this is critical point.

Thank you!

Answer 1

$det(A-\lambda I)=(\lambda+3)^2(\lambda-1)(\lambda-5)$. Hence $det(A-\lambda $I) not zero except $\lambda \neq-3,1,5$