A machine is either working (state 1) or not workind (state 2). If it is working one day the probability that it will be broken the next day is 0.1. If it is not working one day the probability that it will be working the next day is 0.8. Let Tn be the state of the machine n days from now. Assume the Markov assumption is satisfied so that Tn is a Markov Chain.

Question

-Find the eigenvalues and eigenvectors of P. And find a formula for P^n.

Answer 1

If I recall Markov's chain, the matrix for the above would be

.9 .1
.8 .2

Is this called the transition matrix ?

The steady state vector, or eigenvector, is found by multiplying an initial probability vector by that transition vector and then using that resulting vector in your next multiplication.
This will eventually converge to a fixed vector, that is

[a b] times
.9 .1
.8 .2 equals [a b]

then .9a + .1b = a
and .8a + .2b = b
also a + b = 1 by the laws of probability.
solving this I got [.5 .5]

hope this helps, haven't done this stuff in years. I think the last time I taught this was over 20 years ago.

Answer 2

I made a fundamental error in my matrix multiplication
my equations should have been
.9a + .8b = a
and .1a + .2b = b

[a b] should have been [8/9 1/9]