Please help me with linear algebra question.

1. Read Exercises 2 and 12 in Section 4.9. They concern a Markov chain with the system matrix P shown below. In these exercises there are three foods and the i, j entry of P is the probability that if an animal chooses food j on the first trial, then it will choose food i on the second trial. Therefore the i, j entry of P- is the probability that if an animal chooses food j on the first trial, it will choose food i on the third trial. (a) The matrix for exercises 2 and 12 is called P and is shown below. Type P^2 to calculate p2 and record: .5 .25 .25 P2 = P= .25 .5 .25 .25 .25 .5 (b) (by hand) Suppose an animal chooses food #1 on the first trial. Use P and P2 to answer the following. Find the probability the animal will: Choose food #2 on the second trial: Choose food #2 on the third trial: Choose food #3 on the third trial: (c) (MATLAB) Type I=eye(3), rref(P-I) to calculate the reduced echelon form of P - 1. Use this to write the general solution x to the system (P - 1)x = 0 . Also, choose a nonzero value for the free variable and write a particular solution w. Record your results: (general solution) (particular solution) rref(P-D)= X = W = (d) Type the following lines to calculate a steady state vector q for P: w = [ (your data) ] (Use your w from above, which satisfies Pw = w; store it as a column) q = w/sum(w) (Divide w by the sum of its components) Record the result: q = (e) Explain why q is a probability vector. In addition, type sum(q), P*q to verify that q satisfies Pq = q. (Copy and paste the output to show the verification). (f) Explain why P is a regular stochastic matrix, and why q must be its unique steady-state vector. (Use the definition of a regular stochastic matrix and Theorem 18.)