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UNIVERSITY OF MALTA FACULTY OF INFORMATION & COMMUNICATION TECHNOLOGY DEPARTMENT OF ARTICIAL INTELLIGENCE & DEPARTMENT OF MATHEMATICS B.Sc. (Hons.) ICT Evening Degree / Diploma in ICT May / June 2008 Assessment Session 4th June 2008 MAT1092: Mathematical Methods for ICT 18:00 – 19:30 This paper contains two sections. You are to attempt two questions from each section. Section A – Matrices 1) a. If A is the matrix formed from the left-hand side of the equations, while B is the matrix formed from their answers, determine: i. A.B 2x + 9y – 5z = -7 z – 2y = 2 2z – x – 3y = 3 ii. At iii. A-1 b. [2 marks] [2 marks] [2 marks] iv. Bt [2 marks] v. Bt.A-1 [2 marks] vi. |A| [5 marks] vii. The values of x, y, and z using A-1 [5 marks] viii. Confirm the values in (vii). by employing Cramer’s Rule [5 marks] Consider the matrix 5 -3 3 2 6 -3 4 2 1 and determine: i. the Eigenvalues [9 marks] ii. the corresponding Eigenvectors [12 marks] iii. whether it is diagonizable or not (show working) [4 marks] Total marks for question one: 50 marks Page 1 of 3 2) If M = 1 2 1 6 -1 0 -1 -2 -1 , determine: i. M.Mt [3 marks] ii. |M| [7 marks] iii. the matrix N, where N = Mt.M – 2I, where I is the identity matrix [10 marks] iv. the rank of M-1, Mt, M.Mt, Mt.M, and N [10 marks] v. the eigenvalues and eigenvectors of M [10 marks] vi. Hence, or otherwise, determine M5 [10 marks] Total marks for question two: 50 marks 3) a. Q= ½ 2 - /2 ½ ½ 2 /2 ½ 2 /2 0 -2/2 i. Work out Q.Qt. [5 marks] ii. Hence determine Q-1. [5 marks] iii. Find |Q|. [5 marks] b. If A = 1 2 -1 2 1 1 5 4 1 and B = 1 -1 0 -1 1 1 -1 1 2 i. Calculate A.B [7 marks] ii. Determine the eigenvalues of A iii. Show that two of the eigenvectors of A are iv. Find the third eigenvector 1 -1 -1 and v. Is A diagonalizable? And if so, work out the value of A7 -1 1 1 [7 marks] [7 marks] [7 marks] [7 marks] Total marks for question three: 50 marks Page 2 of 3 Section B – O.D.E. 4) Solve the following differential equations: a. (x + y – 2) dx + (x – y + 4) dy = 0 [15 marks] b. dy [15 marks] c. y” + 3y’ – 4y = 18e2x [20 marks] /dx + 2y/x = -2x4/3 Total marks for question six: 50 marks 5) Prove that the following FODE are not exact, determine their Integrating factor, and eventually prove that they can be made exact: a. 2xy.y’ + x2 = -3y2 [15 marks] b. x.y’ + y +x2y = 0 [15 marks] c. y’ [20 marks] /x + 2y = -1 Total marks for question four: 50 marks 6) Find the general solutions to the following SODEs: a. d2y/dx2 – /dx – 12y = 14sin2x + 18cos2x b. d2y/dx2 – 3 dy/dx + 2y = 3e-x [15 marks] c. y” – 2y’ + y = 2x2 + 2x -7 [20 marks] dy [15 marks] Total marks for question five: 50 marks Page 3 of 3