You don't need to search for complicated counterexamples; just consider:

• the matrix with all elements equal to 1
• the matrix with ones in the upper triangular part and 0 in the lower triangular part.

You don't need to search for complicated counterexamples; just consider the matrix with all elements equal to 1.

[EDIT: removed a second counterexample after a comment pointed out it was reducible. If you want an example with all distinct eigenvalues, you can take the cyclic shift matrix.]