H52 - Math 330 Homework 5.2 (Pages 317,318) (8) Find the...

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Unformatted text preview: Math 330 Homework 5.2 (Pages 317,318) (8) Find the characteristic polynomial and the eigenvalues of the matrix A = bracketleftBigg 7- 2 2 3 bracketrightBigg SOLUTION: The eigenvalues λ must satisfy det( A- λI ) = 0: 0 = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 7- λ- 2 2 3- λ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = (7- λ )(3- λ )- (2)(- 2) = λ 2- 3 λ- 7 λ + 21 + 4 ⇒ λ 2- 10 λ + 25 = 0 ⇒ ( λ- 5) 2 = 0 So λ = 5 is the only eigenvalue. 1 (12) Find the characteristic polynomial of the matrix below: A = - 1 0 1- 3 4 1 0 2 SOLUTION: We calculate the determinant by expanding along the 3rd row: | A- λI | = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle- 1- λ 1- 3 4- λ 1 2- λ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = (2- λ ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle- 1- λ- 3 4- λ vextendsingle vextendsingle vextendsingle...
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This note was uploaded on 09/19/2009 for the course MATH 330 taught by Professor Johnson,j during the Spring '08 term at Nevada.

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H52 - Math 330 Homework 5.2 (Pages 317,318) (8) Find the...

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