$(20$ points$)$ Consider the transition diagram below for $6$ websites:

$($a$)$ Can we use the original transition probabilities to calculate the PageRank for these websites? Why? $(5$ points$)$

$($b$)$ What is the Google Matrix M for the following diagram? $(=0\mathrm{.}15)(5$ points$)$

$($c$)$ Compare with the original transition probability matrix, what are the benefits of using Google Matrix? $(5$ points$)$

$($d$)$ How can we solve the PageRank for theses websites? $($Just list the steps and you don$$t need to solve for the results. $(5$ points$)$