This is a homework problem and I have been stuck at it for over an hour. Any hint will be appreciated.
The question states that a town is running a bike sharing program. A bike could be grabbed at a library (L), a coffee shop (C) or a grocery store (G). The question then gives a $3\times 3$ transition matrix $p(i,j)$ with Markov chain property. The question asks about the state distribution on Tuesday, given all three places have the equal number of bikes on Sunday.
Let $(p_1, p_2, p_3)$ denote the fraction of bikes at three stations on Tuesday. My idea is to solve $(p_1, p_2, p_3)p^5 (i,j)=(1/3, 1/3, 1/3)$ and $p_1+p_2 + p_3=1$, where $5$ comes from the difference of number of days between Tuesday and Sunday. However, as you could imagine, the fifth power of a transition matrix is extremely messy. I doubt this approach is correct. Is there any other way to solve this problem?
Thanks in advance!
If you can diagonalize the matrix with
$$ P^{-1}AP = D \\ P^{-1}A^5P= D^5 \\ A^5 = PD^5P^{-1} $$
where power of a diagonal matrix is trivial.