In a certain market, only two brands of lipsticks, A and B, are sold. Given that a lady last purchased lipstick A, there is 80% chance that she would buy the same brand in the next purchase, while if a lady purchased brand B, there is 90% chance that her next purchase would be brand B. Using this information, develop the transition probability matrix. Now, calculate

(i) the probability that if a customer is currently a brand A purchaser, she will purchase brand B two purchases from now;

(ii) the probability that if a customer is a brand B purchaser, she will purchase brand A three periods from now;

(iii) the probability that three periods from now, a customer shall buy brand B, given that the market share of the two brands is as follows: Brand A - 70%, Brand B - 30%;

(iv) the steady state probabilities.