Question 1. JoJo Mobile sells data plans for cellphones. They sell Individual Plans as well as Family Plans. Market analysts at JoJo Mobile have noticed that customers tend to switch between the two plans each year.
• Among the customers with an individual plan, 30% switch to a family plan the next year.
• Among the customers with a family plan, 20% switch to an individual plan in the next year.
This information can be encoded into a 2x2 matrix, as follows:
$\begin{pmatrix} 0.7 & 0.2 \ 0.3 & 0.8 \end{pmatrix}$
(a) Compute the vector $P^{-1} \begin{pmatrix} 400 \ 600 \end{pmatrix}$, and give a physical interpretation of what this vector represents.
(b) The vector $P^3 \begin{pmatrix} 740 \ 650 \end{pmatrix}$, and give a physical interpretation of what this vector represents.
(c) An eigenvector for P with eigenvalue $\lambda = 1$, and give a physical interpretation of what this vector represents.
(d) An eigenvector for P with eigenvalue $\lambda < 1$. Explain why the eigenvector does not have a physical interpretation.
