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Unformatted text preview: Serge Ballif MATH 535 Homework 4 September 21, 2007 1. (Schanuel) Use the fact that a polynomial P ( X ) F [ x ] has at most deg P roots in F 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 F and = det( A ) 6 = 0 , then the inverse of A in Mat n n ( F ) actually lies in Mat n n ( S ) , where S is the subring S = R [1 / ] . b) Show that a square matrix A with entries from Z [ i ] where i 2 =- 1 is invertible iff det A = 1 , i . (a) Cramers 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 A adj is in Mat n n ( S )....
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