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Unformatted text preview: MATH 554 QUALIFYING EXAM AUGUST 2004 J. Lipman Please begin each question 15 on a new sheet. In doing any question, you may assume preceding parts of that question, even if you havent done them. You may quote without proof any standard fact included in the MA554 syllabus. 1. [10 pts] Let M be the matrix 1 2 1 0 1 0 0 0 1 (with entries in the rational field Q ). Find an invertible matrix U such that U- 1 MU has rational canonical form. Hint . First find the canonical matrix C , then solve MU = UC . 2. [10 pts] A matrix A over C has characteristic polynomial X 2 ( X- 4) 6 and minimal polynomial X ( X- 4) 3 . Find all possibilities for the geometric multiplicities of each of the eigenvalues of A . The geometric multiplicity of the eigenvalue is the dimension of the nullspace of A- I . 3. [10 pts] Notation : Let k be a field, and T an indeterminate. M n ( k ) is the ring of n n matrices with entries in k ; I n M n ( k ) is the identity matrix; and...
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