道奇森行列式恒等式

青青子衿

\begin{align*}
\begin{vmatrix}
a_{\overset{\,}{11}}&a_{\overset{\,}{12}}&a_{\overset{\,}{13}}\\
a_{\overset{\,}{21}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&a_{\overset{\,}{23}}\\
a_{\overset{\,}{31}}&a_{\overset{\,}{32}}&a_{\overset{\,}{33}}\\
\end{vmatrix}=
\dfrac{\begin{vmatrix}
\,\begin{vmatrix}
a_{\overset{\,}{11}}&a_{\overset{\,}{12}}\\
a_{\overset{\,}{21}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}\\
\end{vmatrix}
&\begin{vmatrix}
a_{\overset{\,}{12}}&a_{\overset{\,}{13}}\\
\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&a_{\overset{\,}{23}}\\
\end{vmatrix}\,\\
\,\begin{vmatrix}
a_{\overset{\,}{21}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}\\
a_{\overset{\,}{31}}&a_{\overset{\,}{32}}\\
\end{vmatrix}&\begin{vmatrix}
\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&a_{\overset{\,}{23}}\\
a_{\overset{\,}{32}}&a_{\overset{\,}{33}}\\
\end{vmatrix}\,\\
\end{vmatrix}}{\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}}
\end{align*}

\begin{align*}  
\begin{vmatrix}  
a_{\overset{\,}{11}}&a_{\overset{\,}{12}}&a_{\overset{\,}{13}}&a_{\overset{\,}{14}}\\  
a_{\overset{\,}{21}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{23}}}}&a_{\overset{\,}{24}}\\  
a_{\overset{\,}{31}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{32}}}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{33}}}}&a_{\overset{\,}{34}}\\  
a_{\overset{\,}{41}}&a_{\overset{\,}{42}}&a_{\overset{\,}{43}}&a_{\overset{\,}{44}}\\  
\end{vmatrix}=  
\dfrac{\begin{vmatrix}  
\,\begin{vmatrix}  
a_{\overset{\,}{11}}&a_{\overset{\,}{12}}&a_{\overset{\,}{13}}\\  
a_{\overset{\,}{21}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{23}}}}\\  
a_{\overset{\,}{31}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{32}}}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{33}}}}
\end{vmatrix}  
&\begin{vmatrix}  
a_{\overset{\,}{12}}&a_{\overset{\,}{13}}&a_{\overset{\,}{14}}\\  
\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{23}}}}&a_{\overset{\,}{24}}\\  
\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{32}}}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{33}}}}&a_{\overset{\,}{34}}\\  
\end{vmatrix}\,\\  
\,\begin{vmatrix}  
a_{\overset{\,}{21}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{23}}}}\\  
a_{\overset{\,}{31}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{32}}}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{33}}}}\\  
a_{\overset{\,}{41}}&a_{\overset{\,}{42}}&a_{\overset{\,}{43}}\\  
\end{vmatrix}&\begin{vmatrix}  
\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{23}}}}&a_{\overset{\,}{24}}\\  
\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{32}}}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{33}}}}&a_{\overset{\,}{34}}\\  
a_{\overset{\,}{42}}&a_{\overset{\,}{43}}&a_{\overset{\,}{44}}\\  
\end{vmatrix}\,\\  
\end{vmatrix}}{\begin{vmatrix}
\,\,\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{22}}}}&\boxed{\bbox[cyan,2.2pt]{a_{\overset{\,}{23}}}}\,\,\\  
\,\,\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{32}}}}&\boxed{\bbox[yellow,2.2pt]{a_{\overset{\,}{33}}}}\,\,\\  
\end{vmatrix}}  
\end{align*}