$5.$ Given the directed graph below

$**$View image$**$

respond to the following questions:

a$.)$ Compute the stochastic matrix $M$ associated to the random walk on this directed graph.

b$.)$ Compute the modified matrix $M^{*}$ of the Page Ranking algorithm.

c$.)$ Compute the regularization matrix $M_R$ for $p=\frac{1}{2}$

d$.)$ Does a steady$-$state vector $$ exist for the regular matrix, $M_R?$ If so$,$ why?