Math254OldQuiz4Sols

Math254OldQuiz4Sols - QUIZ 4 SOLUTIONS MATH 254 SUMMER 2001...

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QUIZ 4 - SOLUTIONS MATH 254 - SUMMER 2001 - KUNIYUKI CHAPTER 7, 8 1) Find the eigenvalues of A = - È Î Í ˘ ˚ ˙ 1 1 2 4 . (20 points) Find the eigenvalues: Solve l I A - = 0 . l l l l l l l l l l l l l l 0 0 1 1 2 4 0 1 0 1 0 2 4 0 1 1 2 4 0 1 4 1 2 0 5 4 2 0 5 6 0 3 2 0 2 2 È Î Í ˘ ˚ ˙ - - È Î Í ˘ ˚ ˙ = - - - - ( ) - = - - - = - ( ) - ( ) - - ( )( ) = - + + = - + = - ( ) - ( ) = l =3 or l =2 2) Diagonalize the matrix A = - È Î Í ˘ ˚ ˙ 2 2 1 5 by giving matrices P and D such that D P AP = - 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 l 1 3 = : Solve the system 3 I A - [ ] 0 . 3 2 0 2 0 1 3 5 0 0 - - - - ( ) - È Î Í ˘ ˚ ˙
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1 2 1 2 0 0 - - È Î Í ˘ ˚ ˙ R R R 2 1 2 - Æ 1 2 0 0 0 0 - È Î Í ˘ ˚ ˙ x x x x x t 1 2 1 2 2 2 0 2 - = Æ = = Let . x t x t 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 .
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