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homework set 20-Solutions - BILKENT UNIVERSITY Department...

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B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set 1 # 20 U. Mu˘gan July 4, 2008 Homework problems from the 2 nd Edition, SECTION 6.1 3 (3) ) 2 The characteristic eq. p ( λ ) = λ 2 - 7 λ + 10 = 0 . Therefore, the eigenvalues are λ 1 = 2 , λ 2 = 5 . The eigenvectors for λ 1 = 2: 6 v 11 - 6 v 12 = 0 3 v 11 - 3 v 12 = 0 ~v 1 = 1 1 For λ 2 = 5: 3 v 21 - 6 v 22 = 0 3 v 21 - 6 v 22 = 0 ~v 2 = 2 1 6 (6) ) The characteristic eq. p ( λ ) = λ 2 - 5 λ + 6 = ( λ - 2)( λ - 3) = 0 . Therefore, the eigenvalues are λ 1 = 2 , λ 2 = 3 . The eigenvectors for λ 1 = 1: 4 v 11 - 4 v 12 = 0 3 v 11 - 3 v 12 = 0 ~v 1 = 1 1 For λ 2 = 3: 3 v 21 - 4 v 22 = 0 3 v 21 - 4 v 22 = 0 ~v 2 = 4 3 1 I made every effort to avoid the calculation errors and/or typos while I prepared the solution set. You are responsible to check all the solutions and correct the errors if there is any. If you find any errors and/or misprints, please notify me. 2 The number in the parenthesis denotes the problem number in the International Edition of the textbook
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14 (14) ) The characteristic eq.
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