Chapter 9.2, Problem 63E

Rapid transit. A new rapid transit system has just started operating. In the first month of operation, it is found that $25\%$ of commuters are using the system, while $75\%$ still travel by car. The following transition matrix was determined from records of other rapid transit systems:

$\begin{array}{l}\text{ Next month}\\ \begin{array}{cc}Rapidtransit& Car\end{array}\\ \begin{array}{c}\text{Current}\\ \text{month}\end{array}\begin{array}{c}Rapidtransit\\ Car\end{array}\left[\begin{array}{cc}.8& .2\\ .3& .7\end{array}\right]\end{array}$

(A) What is the initial-state matrix?

(B) What percentage of commuters will be using the new system after $1$ month? After $2$ months?

(C) Find the percentage of commuters using each type of transportation after the new system has been in service for a long time.