行列式学习
矩阵A的行列式定义：$det(A)=\sum\limits_{\sigma \in S_n}sgn(\sigma)\prod\limits_{i=1}^na_{i,\sigma(i)}$
这么一长串公式什么意思呢？
$S_n$表示集合$C={1,2,...,n}$上置换的全体，也就是集合$C$到自身双射的全体。
$a_{i,j}$表示矩阵中第$i$行第$j$列的元素。$sgn(\sigma)$表示置换$\sigma \in S_n$的符号差，也就是逆序对的个数。对于任意正整数$n$,$card(S_n)=n!$，所以是个有限次的求和。
主对角线以下（上）的元素都为0的矩阵叫做上（下）三角矩阵，它的行列式的值与对角行列式一样。
感谢图片提供：这里。
行列式的性质：
1.在行列式中，如果有一行（列）的元素都是0，那么行列式的值为0.
2.行列式中，某一行（列）的元素有公因子$k$，则可以提出$k$后算，算完再乘进去。
3.在行列式中，某一行（列）的元素是两数之和，那么可以拆分成两个行列式相加。
4.行列式的两行（列）互换，改变行列式正负符号。
5.行列式中，如果有两行/列对应成比例或相同，那么行列式的值为0。
6.将一行（列）的$k$倍加进另一行里，行列式的值不变。
7.矩阵行列式=矩阵的转置的行列式。
2022年8月09日 17:51
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2022年8月23日 15:19
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