###### Preview Activity4.5.1

Suppose that our rental car company rents from two locations $$P$$ and $$Q\text{.}$$ We find that 80% of the cars rented from location $$P$$ are returned to $$P$$ while the other 20% are returned to $$Q\text{.}$$ For cars rented from location $$Q\text{,}$$ 60% are returned to $$Q$$ and 40% to $$P\text{.}$$

We will use $$P_k$$ and $$Q_k$$ to denote the number of cars at the two locations on day $$k\text{.}$$ The following day, the number of cars at $$P$$ equals 80% of $$P_k$$ and 40% of $$Q_k\text{.}$$ This shows that

\begin{equation*} \begin{aligned} P_{k+1} \amp {}={} 0.8 P_k + 0.4Q_k \\ Q_{k+1} \amp {}={} 0.2 P_k + 0.6Q_k\text{.} \\ \end{aligned} \end{equation*}
1. If we use the vector $$\xvec_k = \twovec{P_k}{Q_k}$$ to represent the distribution of cars on day $$k\text{,}$$ find a matrix $$A$$ such that $$\xvec_{k+1} = A\xvec_k\text{.}$$

2. Find the eigenvalues and associated eigenvectors of $$A\text{.}$$

3. Suppose that there are initially 1500 cars, all of which are at location $$P\text{.}$$ Write the vector $$\xvec_0$$ as a linear combination of eigenvectors of $$A\text{.}$$

4. Write the vectors $$\xvec_k$$ as a linear commbination of eigenvectors of $$A\text{.}$$

5. What happens to the distribution of cars after a long time?

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