Exercise $4.$ Consider the matrix $A=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 0\end{array}\right].$

A is the adjacency matrix of some network. Without sketching the network, write

a $4$ by $4$ matrix whose $(i,j)-$entry is the number of walks of length $4$ from $v_i$ to $v_j.$

$($Hint: in class we have seen that the number of walks of length $l$ from $v_i$ to $v_j$ is

given by $\{$:$[A^l{]}_{ij})$