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Exam3A200 - M340L EXAM 3A 2:00 FALL 2009 Dr Schurle Your...

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M340L EXAM 3A 2:00 FALL, 2009 Dr. Schurle Your name: Your UTEID: Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones, . . . . 1. (10 points) Explain in detail why eigenvalues of a matrix A must be solutions of det( A - λI ) = 0.
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YOUR SCORE: /100 2. (10 points) Explain in detail why a p × p matrix A is diagonalizable exactly when there is a basis for R p consisting of eigenvectors of A . 3. (10 points) Is 6 an eigenvalue of the matrix 7 2 3 1 1 8 4 3 3 6 17 5 - 2 - 4 - 4 8 ? If so, find a basis for its eigenspace. If not, justify your answer.
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4. (10 points) (a) Find the eigenvalues of the matrix A = 1 0 27 2 5 - 8 3 0 1 . (b) Can you tell whether A is diagonalizable, yes or no? Justify your answer. 5. (10 points) The eigenvalues of the matrix A = 11 - 16 4 24 - 45 12 84 - 168 45 are λ = 3 , 3 , 5. Diagonalize A if possible and if not, explain why not.
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6. Let V be a vector space with basis B = { b 1 , b 2 } . Suppose that
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