1. A population growth model: Suppose we have a population of insects divided into three age groups, A, B, and C. Group A consists of insects 0-1 weeks old, group B consists of insects 1-2 weeks old, and group C consists of 2-3 weeks old. Suppose the groups have Ak, Bk, and Ck insects at the end of the kth week. Assume the following three conditions apply to this population: i. Only 10% of age group A survive into the next week and enter age group B. Hence Bk+1 = 11/10/ - Ak ii. Only 40% of age group B survive into the next week and enter age group C. So Ck+1 = Bk. iii. Each insect from group A has offspring, each insect from group B has 4 offspring, and each insect from group C has 5 offspring. Note: In week k +1 the insects in group A are the offspring of all the insects in week k. (a) (2 marks) Use condition (iii) to write an equation for Ak+1 in terms of the population numbers from the previous week, Ak, Bk, Ck. (b) (5 marks) If the insect population starts out with 1000 from each age group, how many insects are in each group at the end of the third week? Fully justify your answers and show all steps-computations can be done in MATLAB. Hint: Consider the column vector [Ao Bo = [Co] [1000] 1000 1000 as the initial population and use the equations in part (i), (ii), and (iii) to set up a matrix equation: Ak+1 Bk+1 Ak = = K | Bk Ck+1 Ck where K is the coefficient matrix for your system of equations found in parts (i), (ii), and (iii). 2. Consider 1 2 -3 0 1 1 -2 A 2 1 3 4 -3 2 2 -8 (a) (1 mark) Enter A on your script. (b) (2 marks) Use MATLAB to compute the matrix polynomial P=A5-3A3 + 7A - 41 where I is the 4 x 4 identity matrix. 3. Let B = 120 17 OLATO (a) (1 mark) Enter B on your script. (b) (2 marks) Use matrix powers in MATLAB to explain the behavior of Bk as k increases indefinitely; that is, as k∞. Write your explanation of your answer as a comment in your MATLAB script. Hint: Experiment and choose large values of k. 4. Let 1 -1 1 C= 2 3 0 0 2 -1 T 2-15 7 8 and D = 4 0 -30 1 3 5 -7 2 1 (a) (2 marks) Enter matrix C and D on your script in MATLAB. (b) (2 marks) Use the matrix inverse command in MATLAB to solve the matrix equa- tion CX = D. =