Activity 4.1.3

Let's consider an example that illustrates how we can put these ideas to use.

Suppose that we work for a car rental company that has two locations, $P$ and $Q$. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. After doing some market research, we determine:

• 80% of the cars rented at location $P$ are returned to $P$ and 20% are returned to $Q$.

• 40% of the cars rented at location $Q$ are returned to $Q$ and 60% are returned to $P$.

1. Suppose that there are 1000 cars at location $P$ and no cars at location $Q$ on Monday morning. How many cars are there are locations $P$ and $Q$ at the end of the day on Monday?

2. How many are at locations $P$ and $Q$ at end of the day on Tuesday?

3. If we let $P_k$ and $Q_k$ be the number of cars at locations $P$ and $Q$, respectively, at the end of day $k$, we then have $$\begin{aligned} P_{k+1}& {}={} 0.8P_k + 0.6Q_k \\ Q_{k+1}& {}={} 0.2P_k + 0.4Q_k\text{.} \\ \end{aligned}$$ We can write the vector $\xvec_k = \twovec{P_k}{Q_k}$ to reflect the number of cars at the two locations at the end of day $k$, which says that $$\xvec_{k+1} = \left[\begin{array}{rr} 0.8 & 0.6 \\ 0.2 & 0.4 \\ \end{array}\right] \xvec_k$$ or $\xvec_{k+1} = A\xvec_k$ where $A=\left[\begin{array}{rr}0.8 & 0.6 \\ 0.2 & 0.4 \end{array}\right]$.

   Suppose that $$\vvec_1 = \twovec{3}{1}, \qquad \vvec_2 = \twovec{-1}{1}$$. Compute $A\vvec_1$ and $A\vvec_2$ to demonstrate that $\vvec_1$ and $\vvec_2$ are eigenvectors of $A$. What are the associated eigenvalues $\lambda_1$ and $\lambda_2$?

4. We said that 1000 cars are initially at location $P$ and none at location $Q$. This means that the initial vector describing the number of cars is $\xvec_0 = \ctwovec{1000}{0}$. Write $\xvec_0$ as a linear combination of $\vvec_1$ and $\vvec_2$.

5. Remember that $\vvec_1$ and $\vvec_2$ are eigenvectors of $A$. Use the linearity of matrix multiplication to write the vector $\xvec_1 = A\xvec_0$, describing the number of cars at the two locations at the end of the first day, as a linear combination of $\vvec_1$ and $\vvec_2$.

6. Write the vector $\xvec_2 = A\xvec_1$ as a linear combination of $\vvec_1$ and $\vvec_2$. Then write the next few vectors as linear combinations of $\vvec_1$ and $\vvec_2$:

   1. $\xvec_3 = A\xvec_2$.

   2. $\xvec_4 = A\xvec_3$.

   3. $\xvec_5 = A\xvec_4$.

   4. $\xvec_6 = A\xvec_5$.

7. What will happen to the number of cars at the two locations after a very long time? Explain how writing $\xvec_0$ as a linear combination of eigenvectors helps you determine the long-term behavior.