一個四階行列式的計算

2018.04.09

\[\det A=\left| \begin{matrix} x& y& z& w\\ y& x& w& z\\ z& w& x& y\\ w& z& y& x\\ \end{matrix} \right| \]

\[=\left( x+y+z+w \right) \left| \begin{matrix} 1& y& z& w\\ 1& x& w& z\\ 1& w& x& y\\ 1& z& y& x\\ \end{matrix} \right| \]

\[=\left( x+y+z+w \right) \left| \begin{matrix} 1& y& z& w\\ 0& x-y& w-z& z-w\\ 0& w-y& x-z& y-w\\ 0& z-y& y-z& x-w\\ \end{matrix} \right| \]

\[=\left( x+y+z+w \right) \left| \begin{matrix} x-y& w-z& z-w\\ w-y& x-z& y-w\\ z-y& y-z& x-w\\ \end{matrix} \right| \]

\[=\left( x+y+z+w \right) \left| \begin{matrix} x+w-y-z& w-z& 0\\ x+w-y-z& x-z& x+y-w-z\\ 0& y-z& x+y-w-z\\ \end{matrix} \right| \]

\[=\left( x+y+z+w \right) \left( x+w-y-z \right) \left( x+y-w-z \right) \left( x+z-w-y \right) \]