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 ( λ ) = Det ( λI- A ) (5 points). • Find the roots of f ( λ ) (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: • λI- A = λ- 1 2 2 λ- 3 Det ( λI- A ) = ( λ- 1)( λ- 3)- 4 = λ 2- 4 λ- 1 • λ = 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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## This note was uploaded on 03/18/2008 for the course EE 141 taught by Professor Balakrishnan during the Fall '07 term at UCLA.

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20075ee141_1_hw4_solutions - EE141 HW 4 Principles of...

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