Connectivity Matrix Consider the diagram below. Each node represents a webpage, and line segments represent hyperlinks that connect them. For example, the single directional arrow from $A_3$ to ${}_6$ indicates there is a hyperlink on page $A_3$ that takes the user to page $A_6.$ Similarly, double$-$sided arrow between $A_1$ and $A_2$ indicates there is a hyperlink on page $A_1$ to page $A_2,$ and there is a hyperlink on page $A_2$ to page $A_1.$

We can create a matrix $C$ to represent this network. We put a $1$ in the $($ij$)-$th entry if $A_i$ is connected to $A_j$; we put $0$ otherwise.

$($a$)$ Build the connectivity matrix $C$ for the network given in the figure.

$($b$)$ If $C$ indicates traveling from one webpage to another by following a single hyperlink, then $C^2=CC$ indicates traveling from one webpage to another by following two hyperlinks. Calculate $C^2$ and determine $(C^2{)}_{3,4}.$ What does $(C^2{)}_{3,4}$ represent?

$($c$)$ A sequence of $n$ hyperlinks moving from one page to another is called a walk of length $n.$ The $($ij$)-$th entry of $C+C^2+C^3$ is the number of walks length at most $3$ from $A_i$ to $A_j.$ Calculate $C+C^2+C^3$ and $(C+C^2+C^3{)}_{1,6}.$ What does $(C+C^2+C^3{)}_{1,6}$ represent?