Math254OldQuiz4Sols

# Math254OldQuiz4Sols - QUIZ 4 - SOLUTIONS MATH 254 - SUMMER...

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QUIZ 4 - SOLUTIONS MATH 254 - SUMMER 2001 - KUNIYUKI CHAPTER 7, 8 1) Find the eigenvalues of A = - È Î Í ˘ ˚ ˙ 11 24 . (20 points) Find the eigenvalues: Solve l IA -= 0 . ll 0 0 0 10 1 02 4 0 0 141 2 0 54 2 0 56 0 32 0 2 2 È Î Í ˘ ˚ ˙ - - È Î Í ˘ ˚ ˙ = -- () - = - = - - ( ) = -+ + = = - - = =3 or =2 2) Diagonalize the matrix A = - È Î Í ˘ ˚ ˙ 22 15 by giving matrices P and D such that DPA P = - 1 , where D is diagonal. You do not have to give P - 1 . Hint: The eigenvalues of this matrix are 3 and 4. (30 points) Find two linearly independent eigenvectors of A : Since we have n = 2 distinct real eigenvalues, A is guaranteed to be diagonalizable. Find an eigenvector for 1 3 = : Solve the system 3 - [] 0 . 32 02 01 3 5 0 0 - È Î Í ˘ ˚ ˙

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12 0 0 - - È Î Í ˘ ˚ ˙ RR R 21 2 -A 00 0 0 - È Î Í ˘ ˚ ˙ xx xt 1 2 2 20 2 -= A = = Let . 1 2 2 = = Ï Ì Ó x = È Î Í ˘ ˚ ˙ = È Î Í ˘ ˚ ˙ x x t 1 2 2 1 Let the eigenvector p 1 = È Î Í ˘ ˚ ˙ 2 1 . Find an eigenvector for l 2 4 = : Solve the system 4 IA - [] 0 . 42 02 01 4 5 0 0 -- () - È Î Í ˘ ˚ ˙ 22 11 0 0 - - È Î Í ˘ ˚ ˙ R 2 1 2 0 0 - È Î Í ˘ ˚ ˙ 1 2 A 0 0 - È Î Í ˘ ˚ ˙ 2 0 A= = Let .
xt 1 2 = = Ï Ì Ó x = È Î Í ˘ ˚ ˙

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## This note was uploaded on 09/08/2011 for the course MATH 254 taught by Professor Howard during the Spring '09 term at Mesa CC.

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Math254OldQuiz4Sols - QUIZ 4 - SOLUTIONS MATH 254 - SUMMER...

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