1. ## Transition matrix

A faculty has three kinds of students;
- mathematics students = S1
- physicists = S2
- and those who do not study anything at all = S3

We call the number of students in each group S1, S2 and S3.

Every year 10% of the mathematicians and 20% of the physicists takes a break and stop studying (but remains on the faculty).

20% moves from "those who do not study" to mathematicians and 40 % from "those who do not study" to physicists.

Additionally, some physicists finds out that mathematics is very important, so 20% of the physicists is moves to the mathematics.

We can describe this using a transition matrix, A.

a) Determine A

This is what i have done so far;
I think they want the eigenvectors of a 3x3 matrix...

What should i do next?

2. Since $\displaystyle S_{k+1}=A^kS_0$, you want to find $\displaystyle P,D$ such as $\displaystyle A=PDP^{-1}\Rightarrow A^k=PD^kP^{-1}$, where $\displaystyle D$ is a diagonal matrix.

Hence, $\displaystyle D^k=\left(d^k_{i,i}\right)$ and you can compute $\displaystyle S_{k+1}.$