554-HW-6q - F n and A is the matrix of T in this ordered...

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6) List explicitly the six permutations of degree 3, state which are odd and which are even, and use this to give the complete formula for the determinant of a 3 × 3 matrix. 6 Using the traditional notation for a permutation, let σ = (1234) and τ = (132). (a) Are σ and τ odd or even? (b) Find στ and τσ . 6 Let A F n × n be a triangular matrix. Prove its determinant is equal to the product of its diagonal entries. 6 Show that det( xI - A ) is a 3rd degree monic polynomial. 6 Let T be a linear operator defined on F n and define D T ( a 1 , ··· ,a n ) = det( Ta 1 , ··· ,Ta n ) (a) Show D T is n -linear and alternating. (b) If we let c = det( 1 , ··· ,T± n ) then show for any n vectors v 1 , ··· ,v n we have det( Tv 1 , ··· ,Tv n ) = c det( v 1 , ··· ,v n ). (c) If B is any ordered basis for
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Unformatted text preview: F n and A is the matrix of T in this ordered basis, show that det A = c . (d) What is a reasonable name for the scalar c 6 If A is n n over C for odd n , and A T =-A , i.e. A is skew symmetric, then show det A = 0. 6 If A is orthogonal show det A = 1. 6 Let A be n n over C and suppose that it is unitary, i.e. AA * = I n . Show that | det( A ) | = 1. 6 Let A F n n . Prove that there are at most n distinct scalars c in F such that det( cI-A ) = 0 . 6 Let A and B be n n over F . Show that if A is invertible there are at most n scalars c such that cA + B is not invertible. 1...
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This document was uploaded on 01/25/2012.

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