real spectrum of an almost symmetric stochastic matrix

Question

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each row is 1). I wish to prove that all the eigenvalues of $M$ are real. Do you think this is true?

Answer 1

No. The eigenvalues of $ \begin{pmatrix} 1/2 & 1/3 & 1/6 \\ 1/6 & 1/2 & 1/3 \\ 1/3 & 1/6 & 1/2 \\ \end{pmatrix}$ are $1,\ 1/4+i\sqrt{3}/12,\ \text{and } 1/4-i\sqrt{3}/12.$