Consider the following matrix $$ A=\left[ \begin {array}{cccc} 1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0 \end {array} \right]. $$ Assume that $B=A^k$ for some positive integer $k$.

*My Question:*

How to prove there is no $k$ such that all entries of $B$ are odd numbers.

In terminology of graph theory, we should prove there is no positive integer $k$ such that the numbers of walks of length $k$ from any vertex $v_i$ to $v_j$ with $1\leq i,j \leq 4$ are odd numbers.

Thanks for any suggestions.