The states are dropped out s1 graduated s2 is a third-year

Question

For the Markov chain of Exercise, solve the equations (I - Q)Φ = 1 to obtain the fundamental matrix Φ = (I - Q) ^{- 1} .

Exercise:-

The following problem is based upon one in Kemeny and Snell (1960). In each year of a three -year degree course, a university student has probability p of not returning the following year, probability q ofhaving to repeat the year and probability r of passing (p +q +r = 1). The states are: dropped out (s₁); graduated (S₂), is a third-year student (S₃), is a secondyear student (S₄); and is a first-year student (S₅);

Find the transition matrix P and the matrices Q and R. (Note that this is a random walk with absorbing barriers.)