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复旦大学谢启鸿高等代数每周一题[2021A02]参考解答

2021-10-16 22:11 作者:CharlesMa0606  | 我要投稿

本文是本人给出的2021年复旦大学谢启鸿高等代数的每周一题[问题2021A01]的解答

题目来自于复旦大学谢启鸿教授在他的博客提供的每周一题练习

（链接：https://www.cnblogs.com/torsor/p/15329047.html）

本文仅供学习交流，如有错误恳请指正！

[问题2021A02]求下列n阶行列式的值：

%5Cleft%7CA%5Cright%7C%3D%5Cleft%7C%5Cbegin%7Bmatrix%7D1%26x_1%26%5Ccdots%26x_1%5E%7Bn-2%7D%26%5Cleft(x_2%2Bx_3%2B%5Ccdots%2Bx_n%5Cright)%5En%5C%5C1%26x_2%26%5Ccdots%26x_2%5E%7Bn-2%7D%26%5Cleft(x_1%2Bx_3%2B%5Ccdots%2Bx_n%5Cright)%5En%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%26%5Cvdots%5C%5C1%26x_n%26%5Ccdots%26x_n%5E%7Bn-2%7D%26%5Cleft(x_1%2Bx_2%2B%5Ccdots%2Bx_%7Bn-1%7D%5Cright)%5En%5C%5C%5Cend%7Bmatrix%7D%5Cright%7C.%0A

解  注意到第k行的最后一列为：

%5Cleft(x_1%2Bx_2%2B%5Ccdots%2B%5Cwidehat%7Bx_k%7D%2B%5Ccdots%2Bx_n%5Cright)%5En%3D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i-x_k%5Cright)%5En

利用二项式定理展开，有：

%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i-x_k%5Cright)%5En%3D%5Cleft(-1%5Cright)%5Enx_k%5En%2BC_n%5E1%5Cleft(-1%5Cright)%5E%7Bn-1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_k%5E%7Bn-1%7D%2B%5Ccdots-C_n%5E%7Bn-1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)%5E%7Bn-1%7Dx_k%2B%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)%5En

于是可以利用前n-1列消去最后一列对应次数的项，有：

%5Cleft%7CA%5Cright%7C%3D%5Cleft%7C%5Cbegin%7Bmatrix%7D1%26x_1%26%5Ccdots%26x_1%5E%7Bn-2%7D%26%5Cleft(-1%5Cright)%5Enx_1%5En%2BC_n%5E1%5Cleft(-1%5Cright)%5E%7Bn-1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_1%5E%7Bn-1%7D%5C%5C1%26x_2%26%5Ccdots%26x_2%5E%7Bn-2%7D%26%5Cleft(-1%5Cright)%5Enx_2%5En%2BC_n%5E1%5Cleft(-1%5Cright)%5E%7Bn-1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_2%5E%7Bn-1%7D%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%26%5Cvdots%5C%5C1%26x_n%26%5Ccdots%26x_n%5E%7Bn-2%7D%26%5Cleft(-1%5Cright)%5Enx_n%5En%2BC_n%5E1%5Cleft(-1%5Cright)%5E%7Bn-1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_n%5E%7Bn-1%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright%7C

%3D%5Cleft(-1%5Cright)%5E%7Bn-1%7D%5Cleft%7C%5Cbegin%7Bmatrix%7D1%26x_1%26%5Ccdots%26x_1%5E%7Bn-2%7D%26-x_1%5En%2BC_n%5E1%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_1%5E%7Bn-1%7D%5C%5C1%26x_2%26%5Ccdots%26x_2%5E%7Bn-2%7D%26-x_2%5En%2BC_n%5E1%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_2%5E%7Bn-1%7D%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%26%5Cvdots%5C%5C1%26x_n%26%5Ccdots%26x_n%5E%7Bn-2%7D%26-x_n%5En%2BC_n%5E1%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)x_n%5E%7Bn-1%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright%7C

利用拆分法，我们有：

%5Cleft%7CA%5Cright%7C%3D%5Cleft(-1%5Cright)%5En%5Cleft%7C%5Cbegin%7Bmatrix%7D1%26x_1%26%5Ccdots%26x_1%5E%7Bn-2%7D%26x_1%5En%5C%5C1%26x_2%26%5Ccdots%26x_2%5E%7Bn-2%7D%26x_2%5En%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%26%5Cvdots%5C%5C1%26x_n%26%5Ccdots%26x_n%5E%7Bn-2%7D%26x_n%5En%5C%5C%5Cend%7Bmatrix%7D%5Cright%7C%2B%5Cleft(-1%5Cright)%5E%7Bn-1%7DC_n%5E1%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)%5Cleft%7C%5Cbegin%7Bmatrix%7D1%26x_1%26%5Ccdots%26x_1%5E%7Bn-2%7D%26x_1%5E%7Bn-1%7D%5C%5C1%26x_2%26%5Ccdots%26x_2%5E%7Bn-2%7D%26x_2%5E%7Bn-1%7D%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%26%5Cvdots%5C%5C1%26x_n%26%5Ccdots%26x_n%5E%7Bn-2%7D%26x_n%5E%7Bn-1%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright%7C

%3D%5Cleft(-1%5Cright)%5En%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)%5Cprod_%7B1%5Cle%20i%3Cj%5Cle%20n%7D%5Cleft(x_j-x_i%5Cright)%2B%5Cleft(-1%5Cright)%5E%7Bn-1%7DC_n%5E1%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)%5Cprod_%7B1%5Cle%20i%3Cj%5Cle%20n%7D%5Cleft(x_j-x_i%5Cright)

%3D%5Cleft(-1%5Cright)%5E%7Bn-1%7D%5Cleft(%5Csum_%7Bi%3D1%7D%5E%7Bn%7Dx_i%5Cright)%5Cprod_%7B1%5Cle%20i%3Cj%5Cle%20n%7D%5Cleft(x_j-x_i%5Cright)%5Cleft(n-1%5Cright).


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