线性代数笔记18--行列式公式、代数余子式

1. 行列式公式推导

二阶行列式推导
$$\begin{array}{rl}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]& =\left[\begin{array}{cc}a& 0\\ c& d\end{array}\right]+\left[\begin{array}{cc}0& b\\ c& d\end{array}\right]\\ & =\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]+\left[\begin{array}{cc}a& 0\\ c& 0\end{array}\right]+\left[\begin{array}{cc}0& b\\ c& 0\end{array}\right]+\left[\begin{array}{cc}0& b\\ 0& d\end{array}\right]\\ & =\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]-\left[\begin{array}{cc}b& 0\\ 0& c\end{array}\right]\\ & =ad-bc\end{array}$$

三阶行列式推导

$$\begin{gathered} \left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]=\left[\begin{array}{ccc}a& 0& 0\\ d& e& f\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}0& b& 0\\ d& e& f\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}0& 0& c\\ d& e& f\\ g& h& i\end{array}\right] \\ =\left[\begin{array}{ccc}a& 0& 0\\ 0& e& 0\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}a& 0& 0\\ 0& 0& f\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}0& b& 0\\ d& 0& 0\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}0& b& 0\\ 0& 0& f\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}0& 0& c\\ d& 0& 0\\ g& h& i\end{array}\right]+\left[\begin{array}{ccc}0& 0& c\\ 0& e& 0\\ g& h& i\end{array}\right] \\ =\left[\begin{array}{ccc}a& 0& 0\\ 0& e& 0\\ 0& 0& i\end{array}\right]+\left[\begin{array}{ccc}a& 0& 0\\ 0& 0& f\\ 0& h& 0\end{array}\right]+\left[\begin{array}{ccc}0& b& 0\\ d& 0& 0\\ 0& 0& i\end{array}\right]+\left[\begin{array}{ccc}0& b& 0\\ 0& 0& f\\ g& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& c\\ d& 0& 0\\ 0& h& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& c\\ 0& e& 0\\ g& 0& 0\end{array}\right] \\ =aei+bfg+cdh-ahf-bdi-ceg \end{gathered}$$

行列式公式
$$\begin{gathered} detA=\sum _{j_1,j_2,j_{3}ispermutaion}\pm a_{1j_1}{a}_{2j_2}...a_{n{j}_n} \\ \forall j_{t_1},j_{t_2}\wedge t_1\ne t_2\Rightarrow j_{t_1}\ne j_{t_2} \end{gathered}$$

即选取的列坐标不重复，构成了一个排列。
所以非0项共有$n!$项。

余子式
$$M_{ij}:方阵去掉i行j列后的方阵的行列式$$
代数余子式
$$A_{ij}:(-1{)}^{i+j}{M}_{ij}$$

方阵行列式：
$$detA=\sum _1^n{A}_{ik},1\le i\le n$$

2. 三对角线矩阵

$$\left[\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 1& 0\\ 0& 1& 1& 1\\ 0& 0& 1& 1\end{array}\right]$$

$|A_1|=1$
$|A_2|=0$
$|A_3|=--1$
$|A_4|=-1$
$|A_5|=-0$
$|A_6|=1$

周期为6

$A_n$的意思是以$1,1$为起始点的向右向下扩展$k$个单位的矩阵。

如
$$A_3=\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 1\end{array}\right]$$