Question

Question 3 (covers Unit 6) - 25 marks . Your answers to part (a) of this question must be shown with your own work . Part (b) and (c) of this question are required you to use Mathcad to compute your solutions. You must submit a copy or attached a copy of your Mathcad solutions to your tutor for marking (a) Consider the following matrix 2 1 -2 A-0 5 0 (i) Find three eigenvalues λ of A. One of these eigenvalue is λ =-1. llence find the corresponding eigenvectors of each λ. 1) Úse the rules given in Unit 6 to find the eigenvalues of (2) 2A3+ A-2 3I where I is the identity matrix. Give your working steps 4 (b) This part of the question is concerned with the numerical calculation of the cigenvalues and cigenvectors of the matrix. MATHCAD 10.12 13.36 -21.35 8.85 165 5.10 9.71-6.43 11.25 1.53-18.17 -3.04 15.95-32.9317.05 0.93 You are required to use the MATHCAD worksheet entitled 'Finding eigenvalues and eigenvectors' associated with Activities 5.1-5.3 in Unit 6 (the given matrix is similar to the matrix (A) in the given worksheet) to investigate the effect of carrying out the corresponding iteration procedure. Take care when you edit the matrix A (i) Use your Mathcad software to compute the eigenvalues and eigenvectors of the 31 matrix A. (ii) Investigate each of following 3 cases using your MATHCAD results. You will find it necessary to try different number of iterations (N) in the worksheet during your investigations, you should reset the value of N to 10 at the start of each part. (1) Direct iteration method, start with eo = [1, 0, 0. Or. (Hints: Set IP-1 and P-0 on page 1 of the worksheet. Try the iteration N -10, 100, 200.) 3 (2) Inverse iteration method. start with eo = [1, 0, 0, 0. (Hints: Sel IP-2 and P-0 on page 1 of the worksheet. Try the iteration N - 200, 1000, 2000.) (3) Modified inverse iteration method. start with eo [1, 0, 0. O. (Hints Set IP-3 and P-15 on page 1 of the worksheet. Try the iteration N -10, 50, 100.) In each case of above Part (ii), comment on: whether the iteration converges to a value with 3 decimal places accuracy; if it does converge: o write down which eigenvalue and eigenvector reached: and 'UTİte the number of iterations required for the convergence If it is not converge or is slow, then suggest a reason why it is not coverage.

Answer 1

2. -12 2 6 -2 1 0

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