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# AnswerGuide7 - Answer Guide 7 Math 427K Unique Number 55160...

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Answer Guide 7 Math 427K: Unique Number 55160 Wednesday, October 19, 2011 1. The Cayley–Hamilton Theorem in linear algebra says that every matrix satisfies its char- acteristic equation. This means that if the eigenvalues of A are solutions to the scalar equation 2 + + c = 0 , then A itself solves the analogous matrix equation aA 2 + bA + cI = 0 0 0 0 . Do not try to prove this! Simply verify that it works for the particular matrix A = 1 9 5 9 . Solution: The characteristic equation of A is 0 = det ( A - λI ) = λ 2 - 10 λ - 36 . So we compute that A 2 - 10 A - 36 I = 1 9 5 9 2 - 10 1 9 5 9 - 36 1 0 0 1 = 46 90 50 126 - 10 90 50 90 - 36 0 0 36 = 0 0 0 0 . ( F ) 2. Find the general solution of X 0 = LX if L = 5 - 4 8 - 7 . Solution: The characteristic polynomial of the matrix is 0 = λ 2 + 2 λ - 3 = ( λ + 3)( λ - 1). An eigenvector associated to the eigenvalue λ = - 3 is 1 2 ; and an eigenvector associated to the eigenvalue λ = 1 is 1 1 . (Any nonzero multiples work as well.) So the general solution is X ( t ) = αe - 3 t 1 2 + βe t 1 1 . ( F ) 1

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3. The mixing example I gave in the lecture introducing systems led to the system X 0 = MX , where M = 1 100 - 1 1 1 - 1 . Find its general solution.
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AnswerGuide7 - Answer Guide 7 Math 427K Unique Number 55160...

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