Help in an exercise on Markov chain

Question

I'm doing an exercise on Markov chain and I'm in doubt if I'm doing it right.

Below is the image in the question and soon after my response. MY QUESTION.

In the first item my answer was as follows

$$0.08333333, 0.2132867, 0.3868007, 0.2226836, 0.145979$$

Where I multiplied the matrix $P$ by $P$ and made $\alpha P^2$,

My doubts are as follows. The sum of my lines is not $1$. And when you tried trying to do item $B$ by doing the matrix $P ^ {100}$ for example this is going up the values ​​pro infinito.

I would like help if I am doing something wrong. Thanks in advance.

Answer 1

You probably missed some number at the time of computing

Answer 2

Your approach is correct, but you’re right to have doubts about your result. If the row sums (and the sum of the elements of the resulting distribution) are not all equal to $1$, then you’ve made a computational error somewhere. With this particular $\alpha$, $\alpha P^2$ is just the last row of $P^2$. The only element that’s incorrect is the first one, so recheck your work for the lower-left element of $P^2$. Most of the first column of $P$ is zero, so there’s not a lot of room for making errors there.

For the second part, there’s no need to diagonalize $P$. From the wording of the problem, you can safely assume that there is an unique steady-state distribution and so as Ian mentioned, you just need a left eigenvector of $1$. That is, solve the equation $\mathbf\pi(P-I)=0$ with the additional condition that the elements of $\mathbf\pi$ all lie in $[0,1]$ and sum to $1$. You can, of course, find any solution and then normalize it to meet this condition.