Is uniqueness of steady-state vector sufficient to regular transition matrix?

Question

If P is a transition matrix, then a steady-state vector for is a probability vector q such that $P\mathrm{q}=\mathrm{q}$.

A transition matrix P is regular if some power $P^k$ contain only strictly positive entries.

We know that the steady-state vector is the eigenvector of P associated with the eignvalue 1.

So if rank(P-I)=Dim(P)-1, then the eigenvector of P which column sum is 1 (steady-state vector) will be unique.

Is uniqueness of steady-state vector sufficient to the regularness of P?

Answer 1

Have already found a counterexample.

$$P_2=\begin{pmatrix} 0 & 0.5 & 0.5 & 0\\ 0.5 & 0 & 0 & 0.5\\ 0.5 & 0 & 0 & 0.5\\ 0 & 0.5 & 0.5 & 0\\ \end{pmatrix}$$

with unique steady-state vector $(0.25,0.25,0.25,0.25)^T$

Answer 2

A simpler example:

A=[0 1; 1 0]

The population between two stats keep swapping. The only steady-state vector is [0.5, 0.5]