MA554_AUG02 - det A = 1 Let A be a 3 × 3 real rotation...

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QUALIFYING EXAMINATION AUGUST 2002 MATH 554 - Prof. Moh You have to show your work and reasonings. (10 points) (1) Let A be an invertible square matrix. Prove that there is a set of elementary matrices E 1 , ..., E k such that E k ...E 1 A is the identity. (10 points) (2) Show that the number of invertible n × n matrices over F p is Q i = n - 1 i =0 ( p n - p i ) where p is a prime number and F p is Z/pZ , the finite field of p elements. 1
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(10 points) (3) Find all linear transformation T : R 2 7→ R 2 which carry the line y=2x to the line y=3x. (20 points) (4) Let A, B be two n × n matrices with AB = 0. Show that rank ( A ) + rank ( B ) n . 2
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(10 points) (5) Find all non-isomorphic commutative group of order 50. (10 points) (6) Show that a matrix A is similar to its transpose A t . 3
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(20 points) (7) An n × n matrix A is said to be a rotation matrix if it is orthogonal and
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Unformatted text preview: det A = 1. Let A be a 3 × 3 real rotation matrix. Show that 1 is an eigen value. (10 points) (8) Find the invariant factors of the following matrix; 1 0 0 0 0 2 0 2 0 4 (10 points) (9) Show that an n × n matrix A is c for some constant c iff it commutes with every n × n matrix B (10 points) (10) Let A be an n × n complex matrix. Suppose that A 2 = A . Show that A is self-adjoint iff A * A = AA * 5 (10 points) (11) Find the Jordan canonical form of the following matrix over the complex numbers C ; 2 0 0 1 2 0 1 1 1 (10 points) (12) Describe all bilinear forms f on R 3 which satisfy f ( α,β ) =-f ( β,α ) for all α,β . 6...
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