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math20d fall 09 stephen young midterm2

math20d fall 09 stephen young midterm2 - Math 20D Exam#2 25...

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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 - λ 3 - 6 2 - 1 - λ 2 4 - 2 4 - λ = ( - 6 - λ ) det - 1 - λ 2 - 2 4 - λ - 3 det 2 2 4 4 - λ + ( - 6) det 2 - 1 - λ 4 - 2 = ( - 6 - λ )( - (1 + λ )(4 - λ ) + 4) - 3(8 - 2 λ - 8) - 6( - 4 + 4 + 4 λ ) = ( - 6 - λ )( - 4 - 3 λ + λ 2 + 4) + 6 λ - 24 λ = λ (( - 6 - λ )( λ - 3) - 18) = λ ( 18 - 3 λ - λ 2 - 18 ) = - λ 2 (3 + λ ) . 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 0 0 0 2 - 1 2 0 0 0 Thus, two linearly independent eigenvectors associated with 0 are 1 0 - 1 1 2 0 . 2. (20 points) The matrix A has the eigenvalues 3 ± 2 i , - 1, and 2. The eigenvector are - 1 - 4 2 0 ± i 0 - 1 0 2 , 2 0 3 0 , and - 2 1 4 - 2 , respectively. Write down the real solution to the differential equation x 0 = A x with x (0) = 0 3 14 - 6 .
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