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+n∑j=2(α1jdet A11+n∑k=2(−1)k+1αkjdet Ak1)xj=0(det A)x1+n∑j=2(n∑k=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=n∑k=1(−1)k+1βk1det Bk1=n∑k=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 ◻
Sunday, July 21, 2013
Common Annihilator Through Matrices (11.4.3)
Dummit and Foote Abstract Algebra, section 11.4, exercise 3:
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Let R be a commutative ring with 1 and let V be an R-module with x1,...,xn∈V. Letting W be the column matrix of these elements, assume that for some A∈Mn×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+n∑j=2(α1jdet A11+n∑k=2(−1)k+1αkjdet Ak1)xj=0(det A)x1+n∑j=2(n∑k=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=n∑k=1(−1)k+1βk1det Bk1=n∑k=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 ◻
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+n∑j=2(α1jdet A11+n∑k=2(−1)k+1αkjdet Ak1)xj=0(det A)x1+n∑j=2(n∑k=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=n∑k=1(−1)k+1βk1det Bk1=n∑k=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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