Exercise $3.$ Consider the matrix $A=\left[\begin{array}{ccccc}0& 1& 0& 0& 1\\ 1& 0& 1& 1& 0\\ 0& 1& 0& 0& 1\\ 0& 1& 0& 0& 1\\ 1& 0& 1& 1& 0\end{array}\right].$ On a piece of paper draw

the network having $A$ as its adjacency matrix. When doing so label the vertices of

the network. Then solve the following problems.

$(1)$ Write down all walks of length $3$ from $v_5$ to $v_1.$

$(2)$ Compute $A^2.$

$(3)$ Compute the $(2,4)-$entry and the $(5,1)-$entry of $A^3.$