Question 2.5 (8 marks) Your spaceship made it through the wormhole and now you've landed on a new planet. You observe that this planet is inhabited by two species, called X and Y, both represented by huge amorphic masses. When you arrive (day 0) you observe that there are :60 cubic metres of X and kg cubic metres of Y. The next day, you observe that the original 230 cubic metres of X turned into gaze cubic metres of X and %x0 cubic metres of Y; moreover, the original k0 cubic metres of Y turned into gko cubic metres of Y and gko cubic metres of X. The day after, the same happens with the $1 and k1 cubic metres of X and Y you assume that this is a general reproduction pattern! You keep on measuring the sizes and write cc,- and k,- for the number of X and Y on day 2'. a) Determine 3:1 and k1 in terms of 3:0 and 160. b) Find a matrix A such that the reproduction pattern is described by (ii-+1) = A (:2) for z' 2 0. 2+1 2' c) Soon the X and Y have become aware of your arrival and they suddenly change their behavior. You observe that their new reproduction pattern is now described by $34.1 _ 37; _ 4/3 2/3 (km1) B (kt) where B (1/3 5/3 . You want to see what happens in the long run. Do the following: (i) Compute the eigenvalues of B. (ii) For each eigenvalue of B nd one eigenvector. (iii) Find a diagonal matrix D and an invertible matrix P such that B = PEP1