In at tropical forest, the populations of falcons and mice are related to each other. The mice eat everything they can while the falcons eat the mice. The population sizes of falcons and mice evolve according to the rule. * At the beginning of 2023, there were 10 falcons and 40 mice.  a) Approximately how many falcons and mice will there be at the beginning of 2068? b) c) Find a diagonalization, A = BDB-1, of the transition matrix A from part (b), where D is a 2x2 diagonal matrix, and B is an invertible 2x2 matrix. d) Use your answer to (c) to estimate the (approximate) numbers of falcons and mice there will be at the beginning of 2058.

] *CountAlg.java - Notepad e Edit Format View Help Ln 28, Col 1 100% Windows (CRLF) UTF-8 1 2 3 Problem 1. For this problem, let A := |0 1 1 2 *WeightedOrderAlg.java - Notepad File Edit Format View Help Ln 39, Col 1 100% Windows (CRLF) UTF-8 (e) Find a diagonal matrix (a matrix whose off-diagonal entries are all zeros), D, such that AP = PD. (f) For a generic matrix A, if AP = PD, then as long as P is invertible, A = PDP-!. Referring to the eigenvectors of that generic matrix A, what is a criterion you could check to make sure that its associated matrix P (whose columns are eigenvectors of A) is invertible? Feel free to use the invertible matrix theorem (the “big theorem" from the final lecture). There's definitely more than one right answer! (g) Is the specific P matrix you found in the third part of this problem invertible? If so, compute its inverse. *CountAlg.java - Notepad File Edit Format View Help | 1:36 PM O Type here to search a Rain... G 4) 4/26/2022 (1

In at tropical forest, the populations of falcons and mice are related toeach other. The mice eat everything they can while the falcons eat the mice. The population sizes of falcons and mice evolve according to the rule. * At the beginning of 2023, there were 10 falcons and 40 mice.  a) Approximately how many falcons and mice will there be at the beginning of 2068? b) c) Find a diagonalization, A = BDB-1, of the transition matrix A from part (b), where D is a 2x2 diagonal matrix, and B is an invertible2x2 matrix. d) Use your answer to (c) to estimate the (approximate) numbersof falcons and mice there will be at the beginning of 2058. e) what restrictions on the initial population sizes ensures the long time survival of both species in the forest