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e4-sample-s - Math 204 Final Exam(practice solutions...

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Math 204. Final Exam (practice, solutions) December 12th, 2010 This is a list of practice problems for the material we covered after exam 3. You should also practice problems from previous exams since the final is cumulative. 1. Find the eigenvalues and eigenvectors for the matrix A = 1 2 2 1 . The characteristic polynomial is p ( t ) = t 2 - 2 t - 3 = ( t - 3)( t + 1). So the eigenvalues are 3 and - 1. For t = - 1. A + I = 2 2 2 2 . So ~x ker( A + I ) if and only if 2 x 1 +2 x 2 = 0. So the eigenspace is the span of the vector 1 - 1 . Now A - 3 I = - 2 2 2 - 2 . And the eigenspace is spanned by the vector 1 1 . Note : A is symmetric, the eigenvalues of A are real, and the eigenvectors are perpendicular. 2. Let p be a polynomial, let A C n × n , let λ be an eigenvalue for the matrix A . Prove the following: (a) p ( λ ) is an eigenvalue of p ( A ). Suppose that A~x = λ~x for some non-zero vector ~x . First we prove that A k ~x = λ k ~x for any natural number k . The proof is by induction. The case k = 1 is the assumption. Now assume the result for k and consider A k +1 ( ~x ) = A k ( A ( ~x )) = A k ( λ~x ) = λA k ( ~x ) = λλ k ~x = λ k +1 ~x. Now let p ( t ) = a 0 + a 1 t + · · · + a m t m . We have, p ( A ) ~x = ( a 0 I + a 1 A + · · · + a m A m ) ~x = a 0 ~x + a 1 A ( ~x ) + · · · + A m ~x ) = a 0 ~x + a 1 λ~x + · · · + a m λ m ~x = ( a 0 + a 1 λ + · · · + a m λ m ) ~x = p ( λ ) ~x
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