hw9solns - Section 5.4 2 a det(A I =-2-5 = 0 1 =-2 2 =-5 1...

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Section 5.4 2. a. det( A - λ I ) = ( - 2 - λ )( - 5 - λ ) = 0 λ 1 = - 2 , λ 2 = - 5 λ 1 = - 2; ( A - λ I ) v 1 = 0 v 1 = 1 0 λ 2 = - 5; ( A - λ I ) v 2 = 0 v 2 = 1 - 1 b. det( A - λ I ) = ( - 4 - λ )( - 1 - λ ) = 0 λ 1 = - 4 , λ 2 = - 1 λ 1 = - 4; ( A - λ I ) v 1 = 0 v 1 = 1 0 λ 2 = 3; ( A - λ I ) v 2 = 0 v 2 = 7 3 c. det( A - λ I ) = (1 - λ )(7 - λ ) = 0 λ 1 = 1 , λ 2 = 7 λ 1 = 1; ( A - λ I ) v 1 = 0 v 1 = 1 0 λ 2 = 7; ( A - λ I ) v 2 = 0 v 2 = - 1 2 3. a. det( A - λ I ) = λ 2 - 4 λ + 3 = 0 λ 1 = 3 , λ 2 = 1 λ 1 = 3; ( A - λ I ) v 1 = 0 v 1 = 1 1 λ 2 = 1; ( A - λ I ) v 2 = 0 v 2 = 1 - 1 b. det( A - λ I ) = λ 2 - 4 = 0 λ 1 = 2 , λ 2 = - 2 λ 1 = 2; ( A - λ I ) v 1 = 0 v 1 = 1 1 λ 2 = - 2; ( A - λ I ) v 2 = 0 v 2 = 1 - 1 c. det( A - λ I ) = λ 2 - 2 λ + 8 = 0 λ 1 = 4 , λ 2 = - 2 λ 1 = 4; ( A - λ I ) v 1 = 0 v 1 = 1 1 λ 2 = - 2; ( A - λ I ) v 2 = 0 v 2 = 1 - 1
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Section 6.1 6. a. Eigenvalues: λ 1 = - 2 , λ 2 = 1 Saddle Eigenvectors: v 1 = 2 1 , v 2 = 1 1 General Solution: x ( t ) = c 1 2 1 e - 2 t + c 2 1 1 e t b. Eigenvalues: λ 1 = 3 - 5 2 , λ 2 = 3 + 5 2 Unstable Eigenvectors: v 1 = 1 - 1 . 618 , v 2 = 1 . 618 1 General Solution: x ( t ) = c 1 1 - 1 . 618 e 0 . 382 t + c 2 1 . 618 1 e 2 . 618 t 7. a. Eigenvalues: λ 1 = 5 , λ 2 = 1 Unstable Eigenvectors: v 1 = 1 1 , v 2 = - 1 3 General Solution:
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