AnswerGuide7 - Answer Guide 7 Math 427K: Unique Number...

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Unformatted text preview: Answer Guide 7 Math 427K: Unique Number 55160 Wednesday, October 19, 2011 1. The CayleyHamilton 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 a 2 + b + 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- I ) = 2- 10 - 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 36 = 0 0 0 0 . ( F ) 2. Find the general solution of X = LX if L = 5- 4 8- 7 . Solution: The characteristic polynomial of the matrix is 0 = 2 + 2 - 3 = ( + 3)( - 1). An eigenvector associated to the eigenvalue =- 3 is 1 2 ; and an eigenvector associated to the eigenvalue = 1 is 1 1 . (Any nonzero multiples work as well.) So the general solution is X ( t ) = e- 3 t 1 2 + e t 1 1 . ( F ) 1 3. The mixing example I gave in the lecture introducing systems led to the system X = MX , where M = 1 100- 1 1 1- 1 ....
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This note was uploaded on 10/24/2011 for the course MATH 427K taught by Professor Delallave during the Spring '11 term at University of Texas at Austin.

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AnswerGuide7 - Answer Guide 7 Math 427K: Unique Number...

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