MATH535_HW4 - Serge Ballif MATH 535 Homework 4 1(Schanuel...

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Serge Ballif MATH 535 Homework 4 September 21, 2007 1. (Schanuel) Use the fact that a polynomial P ( X ) [ x ] has at most deg P roots in to show that the Vandermonde determinant is non-zero when the entries are distinct. Recall that det 1 ... 1 c 1 ... c n . . . . . . c n - 1 1 ... c n - 1 n = Y i<j ( c j - c i ) . If c i 6 = c j for i 6 = j , then c j - c i 6 = 0 for i < j . The product of nonzero elements of a field is nonzero, so the Vandermonde determinant is nonzero. 2. a) Show that if A Mat n × n ( R ) where R is a subring of a field and Δ = det( A ) 6 = 0 , then the inverse of A in Mat n × n ( ) actually lies in Mat n × n ( S ) , where S is the subring S = R [1 / Δ] . b) Show that a square matrix A with entries from [ i ] where i 2 = - 1 is invertible iff det A = ± 1 , ± i . (a) Cramer’s rule states that if A is an invertible n × n matrix, then A - 1 = (det A ) - 1 A adj , where A adj is the matrix with entries a ij = ( - 1) i + j det c A ji . Since each cofactor c A ji is in R , we must have A adj Mat n × n ( R ). Therefore, A - 1 = 1 Δ A adj is in Mat n × n ( S ).
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