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. 



Walks of odd Lengths in a Matrix
This matrix is invertible modulo 2, thus so is each its power, but all-ones matrix is singular.


