文章目录
- [1. 行列式](#1. 行列式)
- [2. 3X3行列式分解](#2. 3X3行列式分解)
- [3. 代数余子式](#3. 代数余子式)
1. 行列式
假设给定一个n行n列的矩阵，根据矩阵元素，算出一个数，那个数就是行列式。
d e t ( A ) = ∣ a 11 a 12 ⋯ a n n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋯ ⋮ a n 1 a n 2 ⋯ a n n ∣ \begin{equation} det(A)=\begin{vmatrix} a_{11}&a_{12}&\cdots&a_{nn}\\\\a_{21}&a_{22}&\cdots&a_{2n}\\\\ \vdots&\vdots&\cdots&\vdots&\\\\a_{n1}&a_{n2}&\cdots&a_{nn} \end{vmatrix} \end{equation} det(A)= a11a21⋮an1a12a22⋮an2⋯⋯⋯⋯anna2n⋮ann
∣ 1 0 0 1 ∣ = 1 ; ∣ 0 1 1 0 ∣ = − 1 ; \begin{equation} \begin{vmatrix}1&0\\\\0&1 \end{vmatrix}=1;\begin{vmatrix}0&1\\\\1&0 \end{vmatrix}=-1; \end{equation} 1001 =1; 0110 =−1;
- 行列式加法分解
∣ a b c d ∣ = ∣ a 0 c d ∣ + ∣ 0 b c d ∣ = ∣ a 0 c d ∣ + ∣ 0 0 c d ∣ + ∣ 0 0 c d ∣ + ∣ 0 b c d ∣ = ∣ a 0 c d ∣ + ∣ 0 b c d ∣ = a d − b c \begin{equation} \begin{vmatrix}a&b\\\\c&d \end{vmatrix}=\begin{vmatrix}a&0\\\\c&d \end{vmatrix}+\begin{vmatrix}0&b\\\\c&d \end{vmatrix}=\begin{vmatrix}a&0\\\\c&d \end{vmatrix}+\begin{vmatrix}0&0\\\\c&d \end{vmatrix}+\begin{vmatrix}0&0\\\\c&d \end{vmatrix}+\begin{vmatrix}0&b\\\\c&d \end{vmatrix}=\begin{vmatrix}a&0\\\\c&d \end{vmatrix}+\begin{vmatrix}0&b\\\\c&d \end{vmatrix}=ad-bc \end{equation} acbd = ac0d + 0cbd = ac0d + 0c0d + 0c0d + 0cbd = ac0d + 0cbd =ad−bc
2. 3X3行列式分解
- 分解后，保证每行和每列中只有一个元素。
∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ = ∣ a 11 0 0 0 a 22 0 0 0 a 33 ∣ + ∣ a 11 0 0 0 0 a 23 0 a 32 0 ∣ + \begin{equation} \begin{vmatrix} a_{11}&a_{12}&a_{13}\\\\ a_{21}&a_{22}&a_{23}\\\\ a_{31}&a_{32}&a_{33} \end{vmatrix}=\begin{vmatrix} a_{11}&0&0\\\\ 0&a_{22}&0\\\\ 0&0&a_{33} \end{vmatrix}+\begin{vmatrix} a_{11}&0&0\\\\ 0&0&a_{23}\\\\ 0&a_{32}&0 \end{vmatrix}+ \end{equation} a11a21a31a12a22a32a13a23a33 = a11000a22000a33 + a110000a320a230 +- ∣ 0 a 12 0 a 21 0 0 0 0 a 33 ∣ + ∣ 0 a 12 0 0 0 a 23 a 31 0 0 ∣ + \begin{equation} +\begin{vmatrix} 0&a_{12}&0\\\\ a_{21}&0&0\\\\ 0&0&a_{33} \end{vmatrix}+\begin{vmatrix} 0&a_{12}&0\\\\ 0&0&a_{23}\\\\ a_{31}&0&0 \end{vmatrix}+ \end{equation} + 0a210a120000a33 + 00a31a12000a230 +
- ∣ 0 0 a 13 a 21 0 0 0 a 32 0 ∣ + ∣ 0 0 a 13 0 a 22 0 a 31 0 0 ∣ \begin{equation} +\begin{vmatrix} 0&0&a_{13}\\\\ a_{21}&0&0\\\\ 0&a_{32}&0 \end{vmatrix}+\begin{vmatrix} 0&0&a_{13}\\\\ 0&a_{22}&0\\\\ a_{31}&0&0 \end{vmatrix} \end{equation} + 0a21000a32a1300 + 00a310a220a1300
d e t ( A ) = a 11 a 22 a 33 − a 11 a 23 a 32 − a 21 a 12 a 33 + a 31 a 12 a 23 + a 21 a 32 a 13 + a 31 a 22 a 13 \begin{equation} det(A)=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{21}a_{12}a_{33}+a_{31}a_{12}a_{23}+a_{21}a_{32}a_{13}+a_{31}a_{22}a_{13} \end{equation} det(A)=a11a22a33−a11a23a32−a21a12a33+a31a12a23+a21a32a13+a31a22a13
- 举例说明
∣ 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 ∣ = 0 \begin{equation} \begin{vmatrix} 0&0&1&1\\\\ 0&1&1&0\\\\ 1&1&0&0\\\\ 1&0&0&1 \end{vmatrix}=0 \end{equation} 0011011011001001 =0
3. 代数余子式
代数余子式的作用是将 n 阶的行列式通过代数余子式降到 n-1 阶上。
∣ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ∣ = a 11 ∣ a 22 a 23 a 32 a 33 ∣ − a 12 ∣ a 21 a 23 a 31 a 33 ∣ + a 13 ∣ a 21 a 23 a 31 a 33 ∣ \begin{equation} \begin{vmatrix} a_{11}&a_{12}&a_{13}\\\\ a_{21}&a_{22}&a_{23}\\\\ a_{31}&a_{32}&a_{33} \end{vmatrix}=a_{11}\begin{vmatrix}a_{22}&a_{23}\\\\a_{32}&a_{33}\end{vmatrix}-a_{12}\begin{vmatrix}a_{21}&a_{23}\\\\a_{31}&a_{33}\end{vmatrix}+a_{13}\begin{vmatrix}a_{21}&a_{23}\\\\a_{31}&a_{33}\end{vmatrix} \end{equation} a11a21a31a12a22a32a13a23a33 =a11 a22a32a23a33 −a12 a21a31a23a33 +a13 a21a31a23a33