1. A woman visits four different places: A, B, C, and D. If, at a given time, she is at point A, then the next hour she will be at point D with 100 percent probability. If she is at point B, then she will be at point A in an hour (with 100 percent probability) If she is at point C, then she will be at point A or B, with equal probabilities (that is, 50 percent each). If she is at point D, then she will be at point B or C with equal probability. a) Write down the stochastic matrix M that describes, given a probability vector, the probability that the woman will be at each of the places during the next hour. (b) Find the steady state vector. (This is a probability vector [positive entries that add up to 11 satisfying Mu- [Hint: you should simply assume that λ-1 is an eigenvalue for M (this is guaranteed for all stochastic matrices) and find the corresponding eigenspace. Divide by a suitable constant to get the steady state vector (c) After many hours pass, what are the probabilities that the woman will be at A? B? C? D? (These are recorded in the steady state vector.)