# Eigenvalues of irreducible matrix.

$$A \in M_{3x3}(\mathbb{R})$$ is irreducible and non-negaive matrix with three eigenvalues $$\lambda_1 , \lambda_2 , \lambda_3$$. Can it be true :

1. $$\lambda_1 = \lambda_2 = \lambda_3= \varrho (A)$$
2. $$\lambda_1 =i \ \lambda_2 =-i \ \lambda_3=3$$
3. $$\lambda_1=2 , \lambda_2=-2 , \lambda_3=1$$
4. $$\lambda_1=2 , \lambda_2=1 , \lambda_3=-1$$

I think, that first can't be true, because it comes from Frobenius theorem.But I don't know what about the others. So, 1-No, 2-Yes, 3-No, 4-Yes.

• $\det \mathbf{A}=\prod_{k=1}^{3} \lambda_{k}$ – dantopa Jan 22 at 23:15