MA554_JAN98

# MA554_JAN98 - Qualifying Examination Math 554 Name January...

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Qualifying Examination Name: Math 554 January, 1998 Answering any question you can use the answers to preceding ones NOTATION. M n ( F )istheseto f n × n matrices with elements in a ﬁeld F . 1. For the matrix A = 110 0 - 1 - 100 - 2 - 221 11 - 10 M 4 ( R ) ﬁnd: a) The rational form R and Jordan canonical form J . [10] b) An invertible matrix S M 4 ( R ) such that S - 1 AS = J .[ 5 ] 2. For A =( a ij ) M n ( F ) with n 3, let A a ij ) M n ( F ) be the matrix in which a ij is the cofactor A ij of a ij . Prove that A †† = det( A ) n - 2 A . [10] 3. For all A, B M n ( F ), set h A,B i =tr( AB ). 1. Prove that h , i is a non-degenerate symmetric bilinear form on M n ( F ). [5] Fix C M n ( F )andset S = { A M n ( F ): AC = CA } . 2. Show that S = { BC - CB : B M } . [10] T is a linear operator on a non-zero ﬁnite dimensional vector space V over F . 4. The matrix of T in some basis of V is equal to λ 000 1 λ 00 01 λ 0 μ .
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