4.19 Transition matrices and Markov chains questions for IB Maths AI

Interpret the value 0.3 in context.

Find the eigenvalues of $\mathit{T}$.

Find an eigenvector for the eigenvalue 1, and hence find the long-term proportion of students who buy lunch from the cafeteria.

Write down the transition matrix $\mathit{T}$ for one year, with the first row and column for FitZone.

Find the number of members of each gym at the start of 2027.

Find the eigenvalues of $\mathit{T}$ and a corresponding eigenvector for each.

Write down a matrix $\mathit{P}$ and a diagonal matrix $\mathit{D}$ such that .

Hence show that the number of FitZone members $n$ years after the start of 2025 is $2000+1000(0.7{)}^n$.

Write down the number of FitZone members in the long term.

Find the first year in which FitZone has fewer than 2100 members at the start of the year.

Write down the transition matrix $\mathit{T}$ such that ${\mathit{s}}_{n+1}=\mathit{T}{\mathit{s}}_n$, where ${\mathit{s}}_n=\left(\begin{array}{c}{a}_n\\ b_n\\ c_n\end{array}\right)$ gives the proportions of customers with Aero, Brio and Cellux after $n$ months.

Ji-ho is currently an Aero customer.

Find the probability that he is with Brio at the end of each of the next two months and with Cellux at the end of the third month.

At the start of January, 50 % of customers are with Aero, 30 % with Brio and 20 % with Cellux.

Find the percentage of customers with each network six months later.

Find the exact long-term proportion of customers with each network.

Interpret your answer to part (d)(i) in context.

Brio launches a loyalty scheme. Brio now keeps a proportion $p$ of its customers each month. Of the customers who leave Brio, one quarter move to Aero and three quarters move to Cellux. The other columns of $\mathit{T}$ do not change.

Write down the new second column of $\mathit{T}$ in terms of $p$.

Brio wants to have the largest long-term share of the three networks.

Find the minimum value of $p$ needed, giving your answer as a percentage correct to the nearest whole number.

Several years later Aero closes, and its customers join the other two networks. Daniela now models monthly switching between Brio and Cellux with the transition matrix

where the first row and column refer to Brio.

Find the eigenvalues of $\mathit{T}$.

Find an eigenvector for each eigenvalue.

Hence write down matrices $\mathit{P}$ and $\mathit{D}$ such that $\mathit{T}=\mathit{P}\mathit{D}{\mathit{P}}^{-1}$, where $\mathit{D}$ is a diagonal matrix.

Immediately after Aero closes, 45 % of customers are with Brio.

Show that the proportion of customers with Brio after $n$ months is $\frac{7}{19}+\frac{31}{380}(0.81{)}^n$.

Find the number of months after Aero closes until Brio's share first falls below 40 %.

State one reason why Daniela's Markov chain model may not be realistic.

Write down a transition matrix $\mathit{T}$ for the movement of people between the two towns in one year.

Show that $\left(\begin{array}{c}1\\ 2\end{array}\right)$ is an eigenvector of $\mathit{T}$, and state the corresponding eigenvalue.

Hence find the long-term population of each town.

Write down the transition matrix $\mathit{T}$, with the first row and column for Lyon.

Find the expected number of cars at each office after 5 days.

Find the characteristic polynomial of $\mathit{T}$, giving your answer in the form ${\lambda }^2+b\lambda +c$.

Hence find the eigenvalues of $\mathit{T}$ and a corresponding eigenvector for each.

Write down matrices $\mathit{P}$ and $\mathit{D}$ such that .

Hence find the number of cars at Lyon in the long term.

Find ${\mathit{P}}^{-1}$.

Hence show that, as $n\to \mathrm{\infty }$, ${\mathit{T}}^n\to \left(\begin{array}{cc}\frac{2}{3}& \frac{2}{3}\\ \frac{1}{3}& \frac{1}{3}\end{array}\right)$.

Interpret your answer to part (b) in context.