Telephone company/Change/Exercise

The telephone companies ${}A,B$ and ${}C$ compete for a market, where the market customers in a year ${}j$ are given by the customers-tuple ${}K_{j}=(a_{j},b_{j},c_{j})$ (where ${}a_{j}$ is the number of customers of ${}A$ in the year ${}j$ etc.). There are customers passing from one provider to another one during a year.

1. The customers of ${}A$ remain for ${}80\%$ with ${}A$, while ${}10\%$ of them goes to ${}B$, and the same percentage goes to ${}C$.
2. The customers of ${}B$ remain for ${}70\%$ with ${}B$, while ${}10\%$ of them goes to ${}A$, and ${}20\%$ goes to ${}C$.
3. The customers of ${}C$ remain for ${}50\%$ with ${}C$, while ${}20\%$ of them goes to ${}A$, and ${}30\%$ goes to ${}B$.

a) Determine the linear map (i.e. the matrix) that expresses the customers-tuple ${}K_{j+1}$ with respect to ${}K_{j}$.

b) Which customers-tuple arises from the customers-tuple ${}(12000,10000,8000)$ within one year?

c) Which customers-tuple arises from the customers-tuple ${}(10000,0,0)$ in four years?

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