[Question 44a05] Fudan Higher Algebra I (Class 14) Monday question (week 7)

[Question 2014a05](1) set \ (x_1, x_2 \ cdots, X_n, X \) to all uncertain elements. \ (S_k = X_1 ^ K + x_2 ^ K + \ cdots + x_n ^ k \, (K \ geq 1) \), \ (s_0 = n \), try to find the value of the following determinant:

\ [| A | =\begin {vmatrix} s_0 & S_1 & \ cdots & S _ {n-1} & 1 \ S_1 & S_2 & \ cdots & s_n & X \\\ vdots & \ vdots \ s_n & S _ {n + 1} & \ cdots & S _ {2n-1} & x ^ n \ end {vmatrix }. \]

(2) set \ (A = (A _ {IJ}) \) to \ (n \) square matrix, and try to find the value of the following determinant:

\ [\ Begin {vmatrix} A _ {11} & A _ {12} & \ cdots & A _ {1N} & \\& \ ddots && & \ ddots & \\& & A _ {11} & A _ {12} & \ cdots & A _ {1N} \ A _ {21} & A _ {22} & \ cdots & A _ {2n} & \\& \ ddots && \ ddots & \\& A _ {21} & A _ {22} & \ cdots & A _ {2n} \ vdots & \ vdots & \ vdots \ A _ {N1} & A _ {N2} & \ cdots & A _ {NN} & \\& \ ddots & \\& & _{ n1} & A _ {N2} & \ cdots & A _ {NN} \ end {vmatrix }. \]

[Question 44a05] Fudan Higher Algebra I (Class 14) Monday question (week 7)