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$A \in M_{3x3}(\mathbb{R})$ 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.

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

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