The calculation of the determinant of the matrix and its source code

维基百科的定义：
行列式(Determinant),记作$det(\mathbf{A})$或者$|\mathbf{A}|$,It is a scalar computed on a matrix.定义如下：
$$det(\mathbf{A})=\sum _{\sigma \in {\mathbf{S}}_n}sgn(\sigma )\prod _{i=1}^n{a}_{i,\sigma (i)}$$

其中,${\mathbf{S}}_n$是集合$\left\{\begin{array}{c}1,2,3,...,n\end{array}\right\}$The whole of the replacement,即集合$\left\{\begin{array}{c}1,2,3,...,n\end{array}\right\}$A one-to-one mapping to itself(双射)的全体;${\sum }_{\sigma \in {\mathbf{S}}_n}$表示对${\mathbf{S}}_n$The sum of all elements,i.e. if each$\sigma \in {\mathbf{S}}_n$出现一次,$sgn(\sigma ){\prod }_{i=1}^n{a}_{i,\sigma (i)}$Just once in the addition;Satisfaction for every one$1\le i,j\le n$的数据对$(i,j)$,$a_{i,j}$都是矩阵$\mathbf{A}$的第$i$行,第$j$列的元素.
$sgn(\sigma )$是$\sigma \in {\mathbf{S}}_n$sign difference,具体说,如果满足$1\le i,j\le n$但是$\sigma (i)>\sigma (j)$an ordered pair of numbers$(i,j)$称为$\sigma$的一个逆序.A sequence has multiple inversions,它的数量,可以定义为$N_{\sigma }$.This sign difference satisfies the following formula:
$$sgn(\sigma )=1;(N_{\sigma }为偶数)sgn(\sigma )=-1;(N_{\sigma }为奇数)$$

In the calculation of daily space geometry,遇到的都是$3\times 3$的矩阵,它可以表示为：

$$\mathbf{A}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$$

通过定义,可以简化为如下：
$$det(\mathbf{A})=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{21}{a}_{32}-a_{13}{a}_{22}{a}_{31}-a_{11}{a}_{23}{a}_{32}-a_{12}{a}_{21}{a}_{33}$$

The code will be updated later.

参考资料：
https://zh.m.wikipedia.org/zh-sg/%E8%A1%8C%E5%88%97%E5%BC%8F