一个四阶行列式的计算
2018.04.09
\[\det A=\left| \begin{matrix}
x& y& z& w\\
y& x& w& z\\
z& w& x& y\\
w& z& y& x\\
\end{matrix} \right|
\]
\[=\left( x+y+z+w \right) \left| \begin{matrix}
1& y& z& w\\
1& x& w& z\\
1& w& x& y\\
1& z& y& x\\
\end{matrix} \right|
\]
\[=\left( x+y+z+w \right) \left| \begin{matrix}
1& y& z& w\\
0& x-y& w-z& z-w\\
0& w-y& x-z& y-w\\
0& z-y& y-z& x-w\\
\end{matrix} \right|
\]
\[=\left( x+y+z+w \right) \left| \begin{matrix}
x-y& w-z& z-w\\
w-y& x-z& y-w\\
z-y& y-z& x-w\\
\end{matrix} \right|
\]
\[=\left( x+y+z+w \right) \left| \begin{matrix}
x+w-y-z& w-z& 0\\
x+w-y-z& x-z& x+y-w-z\\
0& y-z& x+y-w-z\\
\end{matrix} \right|
\]
\[=\left( x+y+z+w \right) \left( x+w-y-z \right) \left( x+y-w-z \right) \left( x+z-w-y \right)
\]
