设齐次线性方程组 $\left\{\begin{array}{l}a_{11} x_1+a_{12} x_2+\cdots+a_{1 n} x_n=0 \\ a_{21} x_1+a_{22} x_2+\cdots+a_{2 n} x_n=0 \\ \cdots \quad \cdots \quad \cdots \quad \cdots \\ a_{n-1,1} x_1+a_{n-2,2} x_2+\cdots+a_{n-1, n} x_n=0\end{array}\right.$

证明：$X _0=\left(x_{1_o}, x_{2_o}, \cdots, X_{n_o}\right)^T$ 为该方程组的解，其中

$$
\begin{aligned}
& x_{1_o}=\left|\begin{array}{cccc}
a_{12} & a_{13} & \cdots & a_{1 n} \\
a_{22} & a_{23} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n-1,2} & a_{n-1,3} & \cdots & a_{n-1, n}
\end{array}\right| \\
& x_{2_o}=-\left|\begin{array}{cccc}
a_{11} & a_{13} & \cdots & a_{1 n} \\
a_{21} & a_{23} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n-1,1} & a_{n-1,3} & \cdots & a_{n-1, n}
\end{array}\right| \\
& , \cdots, x_{n_o}=(-1)^{n+1}\left|\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1, n-1} \\
a_{21} & a_{22} & \cdots & a_{2, n-1} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n-1,1} & a_{n-1,2} & \cdots & a_{n-1, n-1}
\end{array}\right|
\end{aligned}
$$


且若 $X_0 \neq 0$ ，则方程组的任一解可以表示为 $k X_0$ ，其中 $k$ 为常数．