##### 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*}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 \(\xvec_{k+1} = A\xvec_k\text{.}\)

Find the eigenvalues and associated eigenvectors of \(A\text{.}\)

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 \(A\text{.}\)

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

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