12 For a Q matrix representing the transitions between transient states in an absorbing Markov chain, it can be shown that

(I  Q)1  I  Q  Q2      Qn    

a Explain why this expression for (I  Q)1 is plausible.

b Define mij  expected number of periods spent in transient state tj before absorption, given that we begin in state ti.

(Assume that the initial period is spent in state ti.) Explain why mij  (probability that we are in state tj initially)  (probability that we are in state tj after first transition)  (probability that we are in state tj after second transition)     
(probability that we are in state tj after nth transition)    .
c Explain why the probability that we are in state tj initially  ijth entry of the (s  m)  (s  m) identity matrix. Explain why the probability that we are in state tj after nth transition  ijth entry of Qn.
d Now explain why mij  ijth entry of (I  Q)1.