5

5. [Maximum mark: 16]

The drivers of a delivery company can park their vans overnight either at its headquarters or at home.

Urvashi is a driver for the company. If Urvashi has parked her van overnight at headquarters on a given day, the probability that she parks her van at headquarters on the following day is 0.88. If Urvashi has parked her van overnight at her home on a given day, the probability that she parks her van at home on the following day is 0.92.

(a) Write down a transition matrix, $T$, that shows the movement of Urvashi’s van between headquarters and home. [2]

On Monday morning she collected her van from headquarters where it was parked overnight.

(b) Find the probability that Urvashi’s van will be parked at home at the end of the week on Friday evening. [3]

(c) Write down the characteristic polynomial for the matrix $T$. Give your answer in the form $\lambda^{2} + b\lambda + c$. [2]

(d) Calculate eigenvectors for the matrix $T$. [4]

(e) Write down matrices $P$ and $D$ such that $T = PDP^{-1}$, where $D$ is a diagonal matrix. [2]

(f) Hence find the long‑term probability that Urvashi’s van is parked at home. [3]