An Internet Service Provider $($ISP$)$ offers three plans, High Speed $($H$),$ Medium Speed $($M$)$ and Low Speed $($L$).$ Each customer has the option of changing plan

each month, and using past data they have estimated the proportion of customers who change between plans as follows:

$0\mathrm{.}26$ of those on H change to $M,0\mathrm{.}1$ change to L $.$

$0\mathrm{.}11$ of those on M change to $H,0\mathrm{.}09$ change to L $.$

$0\mathrm{.}07$ of those on L change to $H,0\mathrm{.}27$ change to M $.$

All other customers stay on the same plan.

$($a$)$ Construct a transition matrix $A,$ such that if $c_k=(H_k,M_k,L_k)$ describes the state at time $k$ then $c_{k+1}=A{c}_k$

$($b$)$ If at the moment $35\%$ of customers are on plan $H,25\%$ on plan M and $40\%$ on plan L $,$ then what proportion are expected to be on each plan after $1$ month? Enter your answer as proportions $($not percentages$)$ in the form H$,$M$,$L with each number accurate to $2$ decimal places.

$($c$)$ What are the expected proportions after $6$ months?

$($d$)$ Suppose the ISP buys out a competitor and all of the competitors customers are transferred to them and by default are on plan M$.$ These customers can

then change to other plans each month. Assuming that they change according to the same probabilities as the existing customers, what will be the

proportions of these new customers on each of plan after $10$ months. Enter once again in the form H$,$M$,$L to two decimal places.