Question

1.) A chemical species can take on three different forms: A, B and C. After some amount of time passes, the chemical can change between each form in a predictable way. If it is found in form A, it has a 10% chance to convert to B, 80% chance to convert to C and 10% chance to remain as A. From form B, it has a 20% chance to become A, 30% chance to convert to C and a 50% chance to remain as B. Finally, if it is in form C, it converts to A 60% of the time, to B 20% of the time and stays in the form of C 20% of the time. (a - 2 points) If you start with 100 kg of pure form A and 50 kg of pure form B and allow the conversion above to take place, how many kg of form C are you likely to have? kg of Al (b - 2 points) Create a matrix, M, that can be multiplied by any vector, y = kg of B [kg of c] (c - 2 points) Find the Eigenvalues of M using Newton's Method to solve the roots of the characteristic equation. (d - 2 points) There is a composition of ū such that My = 7. What will be the proportion of forms A, B and C at this composition? (Hint: Use one of the Eigenvectors to answer this)

Answer 1

Chemical Species can be an Ihree different forms A,B,C. Explaination in the question can be transformad to weathermatical Equations A = 0.1A +0:16+078( Of af found on form AJ 8.2 A+ 8.5 B+0.3 (Wolff found in form B] anh B = (e) if found in forma 10.6A + 0.2 B +0.2c- a) Given that lookg of pure form A and 50 kg of pure from B -> A=100 B-50 let us substitirte in. Es o and 100 = 0.1(100) +0.1(50) 70-80 50=0.2(100) + 0.5(50) + 0.3c 50 - 10 - 20 to.5c = 160 0.50 = 50+30 c So the above reaction will yield 160 kg of from the Equations of O® & © We can verite it in matrin form A Y AP rool 0.1 0.8 0.2 0.5 0.3 0.6 0.2 0.2 010-8 Marol 0.2 0.5 0.3 0.6 0.2 0.2

C) Find the eigenvalues of m using Newtonia method to solve the roots of the characteristic Epuaslon To find the Eigen values (A-XI) is Scalar and called as Eigenvalue of Matrix A, matrix M . Characteristic Bag det (on - A ) =0 Poi-x 1 08 where Here A is nothing but -4 det 0.2 0.5-0.3 0.2, 0.2-1 -> (6.1 -)[Co-5-2) (-21) – (0240-3). 04 (0-2 (0-2-4)=60-270-4)] +6.8 (0.2802)-0.6.(0-5-1)] -0. by sowing the above Equation -0.547, 0.347 Composition of a such that MP-V , find the proportion of AB, C at this point. Let find the Reigen veetor at d=1 (First quare] 0.1 0.8 0.1-1 6 0.2 0.5-1 0.3 JE) 0.6 0.2 -> -0.94 + 0.1B to 8c ron 0.2 0.5 to 30 Let A=1 0.1 B +0.80 -0.95 - 0.58 to 30 --0.2 B = 1, C=1 above the satisfies Vester Reigen mati .: The proportions of A,B,C is 1:1:1 MP=r