Suppose that there are weather patterns in a city. If it is sunny, there is a 20% chance that it will be rainy the next day. If it is raining, there is a 40% chance that it will be sunny the next day. You must show your work clearly and use concepts you have learned in Linear Algebra, not Elementary Algebra. 1. Explain why =1 is an eigenvalue of any stochastic matrix. (solution) 2. Find two linearly independent eigenvectors of the stochastic matrix. Explain why they are linearly independent without using the fact that they are nonparallel. (solution) 3. Diagonalize the stochastic matrix.