行列式的乘法定理
方阵$A$和方阵$B$,可以在$A$和$B$之间做矩阵的乘法,得到方阵$|AB|$.则我们有
$$|AB|=|A||B|$$
证明:为此我们先证明另一个命题,我们要证明必有
引理:$$|AB^T|=|A||B^T|$$
下面用具体实例在说明该引理的正确性.设
\begin{equation}
A=\begin{pmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}\\
\end{pmatrix}
\end{equation}
设
\begin{equation}
B=\begin{pmatrix}
b_{11}&b_{12}&b_{13}\\
b_{21}&b_{22}&b_{23}\\
b_{31}&b_{32}&b_{33}\\
\end{pmatrix}
\end{equation}
下面我们来看$|A||B^T|$,$|A||B^T|$必定有$3!3!=36$项.下面我们来看其中一项.比方说其中一项是
\begin{equation}
a_{12}a_{21}a_{33}b_{13}b_{21}b_{32}
\end{equation}
该项的符号我们暂且不管它，稍后再考虑.
我们现在来看，其实
\begin{equation}
a_{12}a_{21}a_{33}b_{13}b_{21}b_{32}
\end{equation}在
\begin{equation}
|AB^T|
\end{equation}里也是有出现的！而且是只出现一次的！在哪里有出现呢？我们来看
\begin{equation}
a_{11}b_{31}+a_{12}b_{32}+a_{13}b_{33}
\end{equation}以及
\begin{equation}
a_{21}b_{21}+a_{22}b_{22}+a_{23}b_{23}
\end{equation}以及
\begin{equation}
a_{31}b_{11}+a_{32}b_{12}+a_{33}b_{13}
\end{equation}
将这三者相乘，发现里面也有$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$.而且能拼凑出$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$的也只有这三者.下面我们
来表明，$|A||B^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$和$|AB^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$的符号是相同的.这是一个神秘的事情.因此我将用一种神秘的语言来解释——一种形而上的语言.
SVG code
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根据这幅图，很容易辨别出$|A||B^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$和$|AB^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$的符号是相同的.
而且，容易得到$|AB^T|$中的每一项，$|A||B^T|$中也唯一地存在,且易得它们的符号是相同的.综上,可得
$$|AB^T|=|A||B^T|$$
引理得证.
注:根据引理我们便很容易得知$|AB|=|A||B|$了.
