# 范德蒙行列式的形式（C004）

## 选项

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

[B].   $D_{n}$ $=$ $\left|\begin{array}{ccc} x_{1} & x_{2} & x_{3} \\ 2 x_{1} & 2 x_{2} & 2 x_{3} \\ 3 x_{1} & 3 x_{2} & 3 x_{3} \end{array}\right|$

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

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

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

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

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