math20d fall 09 stephen young midterm2

math20d fall 09 stephen young midterm2 - Math 20D Exam #2...

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Unformatted text preview: Math 20D Exam #2 25 November 2009 1. (15 points) Find all the eigenvalues (with multiplicity) and the eigenvector(s) associated with the largest eigenvalue for the following matrix. - 6 3- 6 2- 1 2 4- 2 4 Solution: det - 6- 3- 6 2- 1- 2 4- 2 4- = (- 6- )det- 1- 2- 2 4- - 3det 2 2 4 4- + (- 6)det 2- 1- 4- 2 = (- 6- )(- (1 + )(4- ) + 4)- 3(8- 2 - 8)- 6(- 4 + 4 + 4 ) = (- 6- )(- 4- 3 + 2 + 4) + 6 - 24 = ((- 6- )( - 3)- 18) = ( 18- 3 - 2- 18 ) =- 2 (3 + ) . Thus the eigenvalues are- 3, 0, and 0, with 0 being the largest. Now reducing the matrix associated with the eigenvalue 0, we get - 6 3- 6 2- 1 2 4- 2 4 2- 1 2 Thus, two linearly independent eigenvectors associated with 0 are 1- 1 1 2 . 2. (20 points) The matrix A has the eigenvalues 3 2 i ,- 1, and 2. The eigenvector are - 1- 4 2 i - 1 2 , 2 3 , and - 2 1 4- 2 , respectively. Write down therespectively....
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This note was uploaded on 01/26/2010 for the course MATH 20D 20D taught by Professor Eggers,john during the Fall '09 term at UCSD.

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math20d fall 09 stephen young midterm2 - Math 20D Exam #2...

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