Open junxnone opened 2 years ago
determinant
det(A)
|A|
$D = \begin{vmatrix} a{11} & a{12} & ... & a{1n} \ a{21} & a{22} & ... & a{2n} \ ... & ... & ... & ... \ a{n1} & a{n2} & ... & a_{nn} \ \end{vmatrix}$
$D2 = \begin{vmatrix} a{11} & a{12} \ a{21} & a_{22} \ \end{vmatrix}$
$D2 = a{11}a{22} - a{12}a_{21}$
$D3 = \begin{vmatrix} a{11} & a{12} & a{13} \ a{21} & a{22} & a{23} \ a{31} & a{32} & a{33} \ \end{vmatrix}$
$\begin{vmatrix} {\color{Red} a{11}} & {\color{Red} a{12}} & {\color{Red} a{13}} \ a{21} & {\color{Red} a{22}} & {\color{Red} a{23}} \ a{31} & a{32} & {\color{Red} a{33}} \ \end{vmatrix}\begin{vmatrix} a{11} & a{12} \ {\color{Red} a{21}} & a{22} \ {\color{Red} a{31}} & {\color{Red} a_{32}} \ \end{vmatrix}$
$\begin{vmatrix} a{11} & a{12} & {\color{Blue} a{13}} \ a{21} & {\color{Blue} a{22}} & {\color{Blue} a{23}} \ {\color{Blue} a{31}} & {\color{Blue} a{32}} & {\color{Blue} a{33}} \ \end{vmatrix}\begin{vmatrix} {\color{Blue} a{11}{\color{Blue} }} & {\color{Blue} a{12}} \ {\color{Blue} a{21}} & a{22} \ a{31} & a_{32} \ \end{vmatrix}$
$D3 = {\color{Red} a{11}a{22}a{33} + a{12}a{23}a{31} + a{13}a{21}a{32}}{\color{Blue} -a{13}a{22}a{31}-a{11}a{23}a{32}-a{12}a{21}a_{33}}$
$D{nn} = \begin{vmatrix} a{11} & a{12} & \cdots & a{1n} \ a{21} & a{22} & \cdots & a{2n} \ \vdots & \vdots & \vdots & \vdots \ a{n1} & a{n2} & \cdots & a{nn} \ \end{vmatrix}$
$D{nn} = (-1)^{1+1}a{11}M{11} + (-1)^{1+2}a{12}M{12} + ... + (-1)^{i+j}a{ij}M{ij} + ... + (-1)^{n+n}a{nn}M_{nn}$
$M{12} = \begin{vmatrix} a{11} & a{12} & \cdots & a{1n} \ {\color{Red} a{21}} & a{22} & {\color{Red} \cdots} & {\color{Red} a{2n}} \ {\color{Red} \vdots} & \vdots & {\color{Red} \vdots} & {\color{Red} \vdots} \ {\color{Red} a{n1}} & a{n2} & {\color{Red} \cdots} & {\color{Red} a{nn}} \ \end{vmatrix} = \begin{vmatrix} a{21} & a{23} & \cdots & a{2n} \ a{31} & a{33} & \cdots & a{3n} \ \vdots & \vdots & \vdots & \vdots \ a{n1} & a{n3} & \cdots & a_{nn} \ \end{vmatrix}$
$M{ij}$ 为 矩阵 删除 $a{ij} 所在行列后剩余的部分组成的矩阵$
$D3 = (-1)^{1+1}a{11}M{11} + (-1)^{1+2}a{12}M{12} + ... + (-1)^{3+1}a{31}M_{31}$
$M{11} = \begin{vmatrix} a{22} & a{23} \ a{32} & a_{33} \ \end{vmatrix}$
$D_4 = \begin{vmatrix} 3 & 1 & 1 & 1 \ 1 & 3 & 1 & 1 \ 1 & 1 & 3 & 1 \ 1 & 1 & 3 & 3 \ \end{vmatrix}$
$\overrightarrow{r1=r1+r2+r3+r4} = \begin{vmatrix} 6 & 6 & 6 & 6 \ 1 & 3 & 1 & 1 \ 1 & 1 & 3 & 1 \ 1 & 1 & 3 & 3 \ \end{vmatrix}$
$\overrightarrow{r1=r1\div 6} = 6\begin{vmatrix} 1 & 1 & 1 & 1 \ 1 & 3 & 1 & 1 \ 1 & 1 & 3 & 1 \ 1 & 1 & 3 & 3 \ \end{vmatrix}$
$\overrightarrow{r2=r2-r1,r3=r3-r1,r4=r4-r1} = 6\begin{vmatrix} 1 & 1 & 1 & 1 \ 0 & 2 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 2 \ \end{vmatrix} $
$= 6 \times (1 \times 2 \times 2 \times 2) = 48$
Determinant 行列式
Reference
Brief
determinant-det(A)-|A|$D = \begin{vmatrix} a{11} & a{12} & ... & a{1n} \ a{21} & a{22} & ... & a{2n} \ ... & ... & ... & ... \ a{n1} & a{n2} & ... & a_{nn} \ \end{vmatrix}$
行列式计算
对角线法
二阶行列式
$D2 = \begin{vmatrix} a{11} & a{12} \ a{21} & a_{22} \ \end{vmatrix}$
$D2 = a{11}a{22} - a{12}a_{21}$
三阶行列式
$D3 = \begin{vmatrix} a{11} & a{12} & a{13} \ a{21} & a{22} & a{23} \ a{31} & a{32} & a{33} \ \end{vmatrix}$
$\begin{vmatrix} {\color{Red} a{11}} & {\color{Red} a{12}} & {\color{Red} a{13}} \ a{21} & {\color{Red} a{22}} & {\color{Red} a{23}} \ a{31} & a{32} & {\color{Red} a{33}} \ \end{vmatrix}\begin{vmatrix} a{11} & a{12} \ {\color{Red} a{21}} & a{22} \ {\color{Red} a{31}} & {\color{Red} a_{32}} \ \end{vmatrix}$
$\begin{vmatrix} a{11} & a{12} & {\color{Blue} a{13}} \ a{21} & {\color{Blue} a{22}} & {\color{Blue} a{23}} \ {\color{Blue} a{31}} & {\color{Blue} a{32}} & {\color{Blue} a{33}} \ \end{vmatrix}\begin{vmatrix} {\color{Blue} a{11}{\color{Blue} }} & {\color{Blue} a{12}} \ {\color{Blue} a{21}} & a{22} \ a{31} & a_{32} \ \end{vmatrix}$
$D3 = {\color{Red} a{11}a{22}a{33} + a{12}a{23}a{31} + a{13}a{21}a{32}}{\color{Blue} -a{13}a{22}a{31}-a{11}a{23}a{32}-a{12}a{21}a_{33}}$
代数余子式法
$D{nn} = \begin{vmatrix} a{11} & a{12} & \cdots & a{1n} \ a{21} & a{22} & \cdots & a{2n} \ \vdots & \vdots & \vdots & \vdots \ a{n1} & a{n2} & \cdots & a{nn} \ \end{vmatrix}$
$D{nn} = (-1)^{1+1}a{11}M{11} + (-1)^{1+2}a{12}M{12} + ... + (-1)^{i+j}a{ij}M{ij} + ... + (-1)^{n+n}a{nn}M_{nn}$
$M{12} = \begin{vmatrix} a{11} & a{12} & \cdots & a{1n} \ {\color{Red} a{21}} & a{22} & {\color{Red} \cdots} & {\color{Red} a{2n}} \ {\color{Red} \vdots} & \vdots & {\color{Red} \vdots} & {\color{Red} \vdots} \ {\color{Red} a{n1}} & a{n2} & {\color{Red} \cdots} & {\color{Red} a{nn}} \ \end{vmatrix} = \begin{vmatrix} a{21} & a{23} & \cdots & a{2n} \ a{31} & a{33} & \cdots & a{3n} \ \vdots & \vdots & \vdots & \vdots \ a{n1} & a{n3} & \cdots & a_{nn} \ \end{vmatrix}$
D3 Examples
$D3 = \begin{vmatrix} a{11} & a{12} & a{13} \ a{21} & a{22} & a{23} \ a{31} & a{32} & a{33} \ \end{vmatrix}$
$D3 = (-1)^{1+1}a{11}M{11} + (-1)^{1+2}a{12}M{12} + ... + (-1)^{3+1}a{31}M_{31}$
$M{11} = \begin{vmatrix} a{22} & a{23} \ a{32} & a_{33} \ \end{vmatrix}$
等价转换法
D4 Examples
$D_4 = \begin{vmatrix} 3 & 1 & 1 & 1 \ 1 & 3 & 1 & 1 \ 1 & 1 & 3 & 1 \ 1 & 1 & 3 & 3 \ \end{vmatrix}$
$\overrightarrow{r1=r1+r2+r3+r4} = \begin{vmatrix} 6 & 6 & 6 & 6 \ 1 & 3 & 1 & 1 \ 1 & 1 & 3 & 1 \ 1 & 1 & 3 & 3 \ \end{vmatrix}$
$\overrightarrow{r1=r1\div 6} = 6\begin{vmatrix} 1 & 1 & 1 & 1 \ 1 & 3 & 1 & 1 \ 1 & 1 & 3 & 1 \ 1 & 1 & 3 & 3 \ \end{vmatrix}$
$\overrightarrow{r2=r2-r1,r3=r3-r1,r4=r4-r1} = 6\begin{vmatrix} 1 & 1 & 1 & 1 \ 0 & 2 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 2 \ \end{vmatrix} $
$= 6 \times (1 \times 2 \times 2 \times 2) = 48$
逆序数法