荒原之梦

$D_{n}$ $=$ $\left|\begin{array}{ccc} 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \end{array}\right|$

范德蒙行列式的通用形式：

$\left|\begin{array}{ccccc} x_{\textcolor{orange}{1}}^{\textcolor{cyan}{0}} & x_{\textcolor{orange}{2}}^{\textcolor{cyan}{0}} & x_{\textcolor{orange}{3}}^{\textcolor{cyan}{0}} & \cdots & x_{\textcolor{orange}{n}}^{\textcolor{cyan}{0}} \\ x_{\textcolor{orange}{1}}^{\textcolor{cyan}{1}} & x_{\textcolor{orange}{2}}^{\textcolor{cyan}{1}} & x_{\textcolor{orange}{3}}^{\textcolor{cyan}{1}} & \cdots & x_{\textcolor{orange}{n}}^{\textcolor{cyan}{1}} \\ x_{\textcolor{orange}{1}}^{\textcolor{cyan}{2}} & x_{\textcolor{orange}{2}}^{\textcolor{cyan}{2}} & x_{\textcolor{orange}{3}}^{\textcolor{cyan}{2}} & \cdots & x_{\textcolor{orange}{n}}^{\textcolor{cyan}{2}} \\ \vdots & \vdots & \vdots & & \vdots \\ x_{\textcolor{orange}{1}}^{\textcolor{cyan}{n-1}} & x_{\textcolor{orange}{2}}^{\textcolor{cyan}{n-1}} & x_{\textcolor{orange}{3}}^{\textcolor{cyan}{n-1}} & \cdots & x_{\textcolor{orange}{n}}^{\textcolor{cyan}{n-1}} \end{array}\right|$ $\Rightarrow$

$\left|\begin{array}{ccccc} 1 & 1 & 1 & \cdots & 1 \\ x_{\textcolor{orange}{1}} & x_{\textcolor{orange}{2}} & x_{\textcolor{orange}{3}} & \cdots & x_{\textcolor{orange}{n}} \\ x_{\textcolor{orange}{1}}^{\textcolor{cyan}{2}} & x_{\textcolor{orange}{2}}^{\textcolor{cyan}{2}} & x_{\textcolor{orange}{3}}^{\textcolor{cyan}{2}} & \cdots & x_{\textcolor{orange}{n}}^{\textcolor{cyan}{2}} \\ \vdots & \vdots & \vdots & & \vdots \\ x_{\textcolor{orange}{1}}^{\textcolor{cyan}{n-1}} & x_{\textcolor{orange}{2}}^{\textcolor{cyan}{n-1}} & x_{\textcolor{orange}{3}}^{\textcolor{cyan}{n-1}} & \cdots & x_{\textcolor{orange}{n}}^{\textcolor{cyan}{n-1}} \end{array}\right|$.

$\left|\begin{array}{cc} \boldsymbol{O} & \boldsymbol{\textcolor{orange}{A}} \\ \boldsymbol{\textcolor{cyan}{B}} & \boldsymbol{C} \end{array}\right|$ $=$ $(-1)^{mn}$ $|\boldsymbol{\textcolor{orange}{A}}||\boldsymbol{\textcolor{cyan}{B}}|$

$\left|\begin{array}{ll} \boldsymbol{C} & \boldsymbol{\textcolor{orange}{A}} \\ \boldsymbol{\textcolor{cyan}{B}} & \boldsymbol{O} \end{array}\right|$ $=$ $(-1)^{mn}$ $|\boldsymbol{\textcolor{orange}{A}}||\boldsymbol{\textcolor{cyan}{B}}|$

$\left|\begin{array}{ll} \boldsymbol{O} & \boldsymbol{\textcolor{orange}{A}} \\ \boldsymbol{\textcolor{cyan}{B}} & \boldsymbol{O} \end{array}\right|$ $=$ $(-1)^{mn}$ $|\boldsymbol{\textcolor{orange}{A}}||\boldsymbol{\textcolor{cyan}{B}}|$

$\left|\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{\textcolor{cyan}{B}} \end{array}\right|$ $=$ $|\boldsymbol{\textcolor{orange}{A}}||\boldsymbol{\textcolor{cyan}{B}}|$

$\left|\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}} \end{array}\right|$ $=$ $|\boldsymbol{\textcolor{orange}{A}}||\boldsymbol{\textcolor{cyan}{B}}|$

$\left|\begin{array}{ll} \boldsymbol{\textcolor{orange}{A}} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{\textcolor{cyan}{B}} \end{array}\right|$ $=$ $|\boldsymbol{\textcolor{orange}{A}}||\boldsymbol{\textcolor{cyan}{B}}|$

$\left|\begin{array}{cccc} 0 & & & \lambda_{1} \\ & & \lambda_{2} & * \\ & \cdots\ & \vdots & \vdots \\ \lambda_{n} & \cdots & * & * \end{array}\right|$ $=$ $(-1)^{\frac{n(n-1)}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

$\left|\begin{array}{cccc} * & \cdots & * & \lambda_{1} \\ * & \cdots & \lambda_{2} & \\ \vdots & \cdots & & \\ \lambda_{n} & & & 0 \end{array}\right|$ $=$ $(-1)^{\frac{n(n-1)}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

$\left|\begin{array}{cccc} 0 & & & \lambda_{1} \\ & & \lambda_{2} & \\ & \cdots & & \\ \lambda_{n} & & & 0 \end{array}\right|$ $=$ $(-1)^{\frac{n(n-1)}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

$\begin{vmatrix} \ast & \ast & \textcolor{orange}{\circ}\\ \ast & \textcolor{orange}{\circ} & \ast\\ \textcolor{orange}{\circ} & \ast & \ast \end{vmatrix}$

$\begin{vmatrix} \textcolor{orange}{\odot} & \ast & \ast\\ \ast & \textcolor{orange}{\odot} & \ast\\ \ast & \ast & \textcolor{orange}{\odot} \end{vmatrix}$

$\left|\begin{array}{cccc} a_{11} & 0 & \cdots & 0 \\ a_{21} & a_{22} & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$

$\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ 0 & a_{22} & \cdots & a_{2 n} \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & a_{n n} \end{array}\right|$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$

$\left|\begin{array}{cccc} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & a_{n n} \end{array}\right|$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$