行列式
$
\left|
\begin{matrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{matrix}
\right|
=\Sigma_{j_{1}j_{2}…j_{n}}(-1)^{\tau(j_{1}j_{2}…j_{n})}a_{1j_{1}}a_{2j_{2}}…a_{nj_{n}}
$
其中,$\tau(j_{1}j_{2}…j_{n})$是自然数1~n得一个排列,为这个排列的逆序数,这样的排列共有n!个