20075ee141_1_hw4_solutions

20075ee141_1_hw4_solutions - EE141 HW 4 Principles of...

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Unformatted text preview: EE141 HW 4 Principles of Feedback Control Instructor: Balakrishnan, A.V. Problem Set 4 1. Let A denote the 2 by 2 matrix: A = 1- 2- 2 3 (1) Find: f ( ) = Det ( I- A ) (5 points). Find the roots of f ( ) (5 points). Calculate f ( A ) (5 points). Let T be nonsingular 2 by 2. Show that f ( T- 1 AT ) = T- 1 f ( A ) T (15 points). Calculate Det ( A 2 ) and show that is equal to ( Det ( A )) 2 (5 points). Solution: I- A = - 1 2 2 - 3 Det ( I- A ) = ( - 1)( - 3)- 4 = 2- 4 - 1 = 4 16 + 4 2 = 2 5 f ( A ) = A 2- 4 A- 1 = 0 2 2 f ( T- 1 AT ) = ( T- 1 AT ) 2- 4 T- 1 AT- I = T- 1 A 2 T- 4 T- 1 AT- T- 1 T = T- 1 ( A 2- 4 A- I ) T = 0 2 2 A 2 = A A = 5- 8- 8 13 Det ( A ) = 1 3- 2 2 =- 1 Det ( A 2 ) = 5 13- 8 8 = 1 = ( Det ( A )) 2 1 2. Let B =- 2 1 1 1 (2) Calculate ( AB ) T and show that is equal to B T A T (5 points)....
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20075ee141_1_hw4_solutions - EE141 HW 4 Principles of...

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