a$)$ Let A be the adjacency matrix of a particular graph with nodes labelled a$,$ b$,$ c$,$ d and the

rows and columns of A corresponding to the nodes in this order. In the graph you may

assume a node cannot connect to itself in a loop.

$($i$)$ You are told

A$5=$

$$

$$

$$

$$

$32293329$

$29182918$

$33293229$

$29182918$

$$

$$

$$

$.$

How many distinct walks are there from node c back to itself using exactly five edges?

Clearly state the underlying theory used to deduce your answer.

$($ii$)$ It is also true that

A$6=$

$$

$$

$$

$$

$91659065$

$65586558$

$90659165$

$65586558$

$$

$$

$$

$.$

Determine A