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MA353_S10_mid2_practice_sol

# MA353_S10_mid2_practice_sol - MA 353 Practice Midterm 2...

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MA 353, Practice Midterm 2 March 29, 2010 Name: This exam consists of 6 pages including this front page. Ground Rules (1) No calculator is allowed. (2) Show your work for every problem unless stated otherwise. A correct an- swer without any justification may not receive the full credit. (3) You may use one 4-by-6 index card, both sides. (4) Show your ID on your desk. Score 1 10 2 10 3 10 4 10 5 10 Total 50 1

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2 1. Let A = 1 2 0 0 1 - 1 0 0 2 . (a) Find all the eigenvalues of A and their geometric and algebraic multi- plicities Answer. Consider det( A - λI 3 ) = 1 - λ 2 0 0 1 - λ - 1 0 0 2 - λ = (1 - λ ) 2 (2 - λ ) = 0 which gives λ = 1 , 2 as the eigenvalues with algebraic multiplicities 2 and 1, respectively. Clearly for λ = 2 the geometric multiplicity is 1 since the algebraic multiplicity is 1. So it remains to compute the geometric multiplicity for λ = 2. Recall that the geometric multiplicity of λ = 1 is dim N ( A - I 3 ). The matrix A - I 3 reduces to 0 1 0 0 0 1 0 0 0 , whose rank is 2, and so the nullity ( i.e. the dimension of the null space) is 1. Hence the geometric multiplicity of
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MA353_S10_mid2_practice_sol - MA 353 Practice Midterm 2...

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