MATH 110 - Spring 2008 - Aaron - Midterm 2 (solution)

MATH 110 - Spring 2008 - Aaron - Midterm 2 (solution) -...

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Math 110 Spring 2008 Midterm 2 Write your name and SID on the front of your blue book. All answers and work should also be written in your blue book. You must JUSTIFY your answers, so show your work. Partial credit will be awarded even if answers are incorrect. No notes, books, or calculators. Good luck! 1. (20 pts.) Let A = 1 3 3 6 4 6 - 3 - 3 - 5 . The characteristic polynomial of A is f ( t ) = - ( t 3 - 12 t - 16). a.(5 pts.) Factor f ( t ). SOLUTION: f ( t ) = - ( t - 4)( t + 2) 2 . b.(15 pts.) Find an invertible matrix Q and a diagonal matrix D such that D = Q - 1 AQ . You do not need to compute Q - 1 . SOLUTION: One possible pair, ( Q, D ), is Q = 1 1 1 2 - 1 0 - 1 0 - 1 and D = 4 0 0 0 - 2 0 0 0 - 2 . 2. (20 pts.) a.(7 pts.) Suppose the characteristic polynomial of A is f ( t ) = t 4 - 1. Use the Cayley-Hamilton Theorem to express A 10 + A 8 as a linear combination of I, A, A 2 , A 3 . SOLUTION: BY the C-H Theorem, A 4 = I . Then A 10 = ( A 4 ) 2 A 2 = IA 2 = A 2 and A 8 = I . Thus A 10 + A 8 = A 2 + I . b.(6 pts.) Suppose T has characteristic polynomial f ( t ) = ( - 1) n [ t n - 2 ( t - λ 2 )( t - λ 3 )], where λ 2 6 = λ 3 and λ i 6 = 0. Suppose further that dim N(
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