Regular matrix and regular stochastic matrix

We know that :

1. A matrix is regular if its determinant is non zero.
2. A stochastic matrix is regular if at a certain power all elements are positive.

Question is how can I make the link between the two definitions, if there are any ? thanks in advance.

1 Answer

Consider this matrix \begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{bmatrix}The first power itself has all elements positive, it is doubly stochastic and still has a zero determinant.