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Sunday, July 21, 2013

Common Annihilator Through Matrices (11.4.3)

Dummit and Foote Abstract Algebra, section 11.4, exercise 3:

MathJax TeX Test Page Let R be a commutative ring with 1 and let V be an R-module with x1,...,xnV. Letting W be the column matrix of these elements, assume that for some AMn×n(R),AW=0Prove (det A)xi=0 for i{1,...,n}.

Proof: This implies a systemα11x1+α12x2+...+α1nxn=0α21x1+α22x2+...+α2nxn=0...αn1x1+αn2x2+...+αnnxn=0In fashion of constructing the cofactor formula for the determinant along the first column, multiply the first row by det A11 and for k{2,...,n} add (1)k+1det Ak1 times the kth row to the first row to obtain(det A)x1+nj=2(α1jdet A11+nk=2(1)k+1αkjdet Ak1)xj=0(det A)x1+nj=2(nk=1(1)k+1αkjdet Ak1)xj=0     ()For each j, let Bj be the matrix A with the first column replaced by the jth column. We see0=det Bj=nk=1(1)k+1βk1det Bk1=nk=1(1)k+1αkjdet Ak1so that () collapses to(det A)x1=0By interchanging arbitrary xi with x1 and letting Ai be the matrix A with its first and ith column interchanged, since this operation negates the determinant and by the argument above we have(det A)xi=0=(det A)xi   

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