Hi

I have added some assignment questions Im working on. I would be most grateful for some feedback and/or guidance in regards to my answers as Im not sure in parts if my working out is correct.

Thanks in advance

1. Let dyl dt = yi = 3y1 - 4y2 dt dy2 = 12 = 2y1 - by2, where y 1 (0) = 1 and y 2 (0) = 3. Find the solution of the system and comment on its stability. [10 marks] Answer: Step 1: write in matrix form y' [y1 A = 3 4 y 1 12 6 y = Ly2 Step 2 : find the eigenvalues and eigenvectors Eigenvalues det (A - XI) = 0 A = [3 2 -6 2 -61 = 0 = (3-X)(-6-1) - (-8)=0 = (-18) -31 + 61 + 12 - (-8)=0 = 12 + 31 - 10 = 0 -10 = ( + 5) =0 (1+-2) =0 = ()1 = -5), (12 = 2)Eigenvector 1: X1 = -5 = 13- (-5) -4 [8 2 -6 - (-5). X v1 = 0, = 2 14 X V 1 = 0 -1 0 R1 = - X R1, followed by: R2 = R2 - R1 = 2v11 - 1v12 = 0 = V11 = V 12 2 = V1 Eigenvector 2: 12 = 2 = 3 -2 -4 = X V2 = 0 2 -6-2 X V2 = 0, - 8 -4 0 OR2-2R1 + R2 0] V22 = 1v21 - 4v22 = 0 = V21 = = V21 = = V21 = V 21 1 V22 = Step 3 : General Solution y = y1 System observations: As the system has opposite eigenvalues(one being positive and the other negative), the system is unstable and is known as saddle.2. At a shopping mall, 80% of the customers come to shop, 15% come to eat, and 5% come to socialize (assume that each customer comes to do only one of these activities). Of those who come to shop, 95% are older than age 20 years; of those who come to eat, 65% are older than 20 years; and of those who come to socialize, 10% are older than 20. (a). What is the probability that a randomly selected customer comes is younger than 20 years of age? Answer: 5: {(age20), (age>20 age20 P(Al = % P(B} = )6 P(A)*P(B) _ 0.90.5 PM) 0.5 = 0.5 = 50% = pm} = (b). If one person is randomly selected, find the conditional probability that the person is there to shop given that the person is 18 years old. Answer: \"U\" 30% of customers come to shop, 95% of the customers are aged older than 20, therefore 5% are aged under 20 Let A = Wet B = age