MATH_Final_Fall_2008_Solution

MATH_Final_Fall_2008 - RIT-Dubai EE 0304-870-32 Mathematics for Engrs I FINAL EXAM(Solution Name INSTRUCTIONS 1 Please box your final answers 2 You

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______________________________________________________________________________ EE 0304-807-32 - Prof. S. Dianat Dept. of Electrical Engineering Fall 2008 Midterm Examination II RIT-Dubai EE 0304-870-32: Mathematics for Engrs I FINAL EXAM (Solution) Name: ____________________________ INSTRUCTIONS : 1. Please “box” your final answers. 2. You must show all work (details, derivations, carefully drawn & labeled graphs, equations, etc.) to receive credit. No details = no credit. All derivations must be clearly described showing each mathematical step. Prob 1: /20 Prob 2: /20 Prob 3: /20 Prob 4: /20 Prob 5 /20 =============== Total: /100
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Name:_________________________________ Fall 2008 EE 0304-807-32 - Prof. S. Dianat Dept. of Electrical Engineering Fall 2008 Midterm Examination II 2 Problem 1: (20 points) a) Let 3 4 1 2 3 2 0 2 2 A => 2 3 1 i i 50 ) det( 3 2 1 A 400 ) 50 ( 8 ) det( 8 ) 2 det( A A b) Let 5 1 3 1 4 2 3 2 1 A 3 2 1 , , x x x are eigenvectors of A => 0 , 2 1 x x 0 , 2 3 x x c) Determine whether the following matrices are (1) positive definite (2) positive semidefinite (3) negative definite (4) negative semidefinite 1 0 0 0 3 2 0 2 2 A Answer: positive definite 2 1 1 1 3 1 1 1 1 B Answer: Negative definite d) Let 1 1 2 1 3 0 0 2 1 1 2 1 1 A . The eigenvalues and eigenvectors of matrix A are 2 1 and 3 2 1 1 1 x , and 1 2 2 x
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Name:_________________________________ Fall 2008 EE 0304-807-32 - Prof. S. Dianat Dept. of Electrical Engineering Fall 2008 Midterm Examination II 3 e) Let A be a n n matrix. Then A and A T have the same eigenvalues Please circle: True False f) Let A be a n n matrix. Then A and A T have the same eigenvectors Please circle: True False g) If A A 2 , then the eigenvalues of A are equal to either 0 or 1. Please circle: True False
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Name:_________________________________ Fall 2008 EE 0304-807-32 - Prof. S. Dianat Dept. of Electrical Engineering Fall 2008 Midterm Examination II 4 Problem 2: (Show all work to receive credit) (20 points) Consider symmetric matrices with entries 1 , , 3 , 2 , 1 n just above and just below the main diagonal. All other entries are zero 0 1 1 0 2 A , 0 2 0 2 0 1 0 1 0 3 A , 0 3 0 0 3 0 2 0 0 2 0 1 0 0 1 0 4 A , 5 A a) Find eigenvalues and eigenvectors of 2 A . (5 pts) b) Find eigenvalues and eigenvectors of 3 A . (5 pts) c) If 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 M Show that (5 pts) 4 4 1 A M A M d) Two eigenvalues of 4 A are approximately 3.65 and 0.822, find the other two eigenvalues (5 pts) a) 1 , 1 0 1 1 1 | | 2 1 2 A I 1 1 1 x
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This note was uploaded on 11/10/2008 for the course ME 0304.870.3 taught by Professor Sohaildainat during the Fall '08 term at RIT.

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MATH_Final_Fall_2008 - RIT-Dubai EE 0304-870-32 Mathematics for Engrs I FINAL EXAM(Solution Name INSTRUCTIONS 1 Please box your final answers 2 You

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