Let $ {\bf A}_n$ be an $ 2n \times 2n$ matrix that is defined as follows

$ $ {\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&0&1&1\ 0&0&\cdots&0&0&0&1&0&0&0\ 0&0&\cdots&0&0&1&1&0&0&0\ 0&0&\cdots&1&0&0&0&0&0&0\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\ 1&1&\cdots&0&0&0&0&0&0&0\ 0&0&\cdots&0&0&0&0&0&1&0\ \end{array} \right). $ $

For instance, the matrix $ {\bf A}_5$ is given by

$ $ {\bf A}_5=\left( \begin{array}{cccccccccc} 0&0&0&0&0&0&0&0&1&1\ 0&0&0&0&0&0&1&0&0&0\ 0&0&0&0&0&0&1&1&0&0\ 0&0&0&0&1&0&0&0&0&0\ 0&0&0&0&1&1&0&0&0&0\ 0&0&1&0&0&0&0&0&0&0\ 0&0&1&1&0&0&0&0&0&0\ 1&0&0&0&0&0&0&0&0&0\ 1&1&0&0&0&0&0&0&0&0\ 0&0&0&0&0&0&0&0&1&0 \end{array} \right). $ $

*My Question:* How to show that the $ (n+2)$ th power of $ {\bf A}_n$ , denoted with $ {\bf A}_n^{n+2}$ , is a positive matrix and matrices $ {\bf A}_n^{i}$ with $ 1\leq i \leq n+1$ are not positive matrices?

For instance, it can be checked that $ {\bf A}_5^{7}$ is a positive matrix and matrices $ {\bf A}_5^{i}$ with $ 1\leq i \leq 6$ are not positive matrices.

I know this question is related with the concept of primitive matrices and may be by considering the values of the eigenvalues of $ {\bf A}_n$ we can obtain an answer. But I would like to find out a combinatoric answer. For example, we can consider $ {\bf A}_n$ as an incidence matrix of a weighted directed graph. Then we should check why there is at least a directed walk between every nod of the graph of length $ n+2$ ? The weighted directed graph of $ {\bf A}_5$ can be drawn in the following form

It follows from the given graph that there is at least a directed walk between every nod of the graph of length $ 7$ . Also, it can be checked that there is no directed walk between the node 4 to the node 2 of length less than seven.

Thanks for any suggestions.