Math 214 - Fall 2022 Week 10 - Homework In class: 0.5 0.1 0.2 (1) Let P be the stochastic matrix P = 0.4 0.2 0.4 and let vo, V1, . . . be an associ- 0.1 0.7 0.4 ated Markov chain. 0.4 (a) If the initial state it vo = 0.3 find v1 and v2. 0.3 (b) Find the steady state vector for P. (2) When mixing lye (sodium hydroxide, NaOH) with hydrocloric acid (HCI) they form salt (NaCl) and water (H20). The conversion is not complete, because when NaCl and H20 molecules collide, they may form NaOH and HCI. It is estimated that this only happens in 5% of the instances. Also, NaOH and HCI react in 90% of the instances that these molecules occur. (a) Draw a state diagram for this chemical reaction (b) Express this in terms of a stochastic matrix (c) Assuming it can be modelled by a Markov chain, find the steady state (equilib rium) for the reaction. (d) What does the steady state say about the reaction? (3) Find a steady state vector for P = 10 . If vo is 0.3 does Uk = Pkvo approach the steady state vector? (4) Find the product i (5) Find the eigenvalues of 3 6 7 (6) Is 4 an eigenvalue of 3 2 7 ? If so, find one eigenvector for the eigenvalue 4. 15 6 4 Hand in: (1) A Markov model for weather. Assume the following state diagram describes the fall weather in Hamilton