All questions please, MATLAB needed for 2nd question

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 he 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 = TAK ii. Only 40% of age group B survive into the next week and enter age group C. So 2 Ck+1 EBr. 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 Ax+1 in terms of the population numbers from the previous week, Ak, B, C (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 A. [1000] 1000 CO 1000) as the initial population and use the equations in part (i), (ii), and (iii) to set up a matrix equation: [ Akti TAR Bk+1 = KB Ck+1] Ck where K is the coefficient matrix for your system of equations found in parts (i), (ii), and (iii) For questions 2, 3, and 4 you need to use MATLAB and submit your pdf of the MATLAB work: Open a MATLAB session, create a script and save it under the name: yourfamilyname_yourfirstname_A_2. Do the following work in your script and save and publish it as a PDF, as shown in MATLAB Demo Part 2. Do not use semicolon to suppress output; let the output be seen on the command window, and in the PDF when it is published. You will need to include the published PDF of all the MATLAB work in your electronic submission in Canvas. 2. Consider 1 2 -3 0 1 1 -2 A = 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 2 where I is the 4 x 4 identity matrix. 3. Let B= (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 +0. 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 [2-15 7 87 C= 2 3 0 and D= 4 0 -3 0 1 0 2 -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 MATLAB command help: To get help with constructing an identity matrix in MATLAB, on your command window type >> doc eye To get help on the power command in MATLAB, on your command window type >> doc To get help on the matrix inverse command in MATLAB, on your command window type >> doc inv