linear algebra

175 3. (15 marks) Here we work with the game from the 2/5 previous problem except there is no reproduction, so the Y population will die out. 3/5 (a) (4 marks) Let x, and y, be the expected number of individuals on nodes X and Y at time n. Find a set of linear recursive equations for x and y at time n+ 1 in terms of their values at time n. Find the matrix of this system and calculate its eigenvalues and eigenvectors. (b) (4 marks) Suppose we begin with x - 100,000 and yo -0. Solve the system of (a). Write separate equations for x, and yo. Report them in as simple a form as you can. (c) (3 marks) Check your formulae from (b) in the following manner. First use the formulae to calculate x5 and y5. Display these formulae in their simplest form and evaluate them. Then put your recursive equations on a spread sheet and use it to tabulate x, and y, for 0sns5. Show that you get the same answer with each method. Include a copy of the table obtained from the spreadsheet listing all values from n = 0 ton = 5. (d) (4 marks) In part (d) of Assignment 9 you were asked to calculate the probability that an individual on node X would make at least one future visit to node Y before it is removed from the board. The answer to that was 1/2. Here, you are to calculate the probability that an individual on node X will make exactly one future visit to node Y before it is removed from the board. By the way there is a problem like this on a recent exam (2018 #7) but the solution there might not be the best way to solve this problem. It's worth thinking about the problem on your own