## 问题

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

## 选项

[A].   $D$ $=$ $\left(x_{2}+x_{1}\right)$ $\cdot$ $\left(x_{3}+x_{1}\right)$ $\cdot$ $\left(x_{3}+x_{2}\right)$

[B].   $D$ $=$ $\left(x_{2}-x_{1}\right)$ $+$ $\left(x_{3}-x_{1}\right)$ $+$ $\left(x_{3}-x_{2}\right)$

[C].   $D$ $=$ $\left(x_{3} – x_{2} – x_{1}\right)$

[D].   $D$ $=$ $\left(x_{2}-x_{1}\right)$ $\cdot$ $\left(x_{3}-x_{1}\right)$

$\left|\begin{array}{ccc} 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \end{array}\right|$ $=$ $\left(x_{2}-x_{1}\right)$ $\cdot$ $\left(x_{3}-x_{1}\right)$ $\cdot$ $\left(x_{3}-x_{2}\right)$

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

$\prod_{1 \leq j < i \leq n}$ $\left(x_{i} – x_{j} \right)$

## 选项

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

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

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

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

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

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

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

## 问题

$\left|\begin{array}{cc} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{C} \end{array}\right|$.

## 选项

[A].   $D$ $=$ $|\boldsymbol{A}|$ $+$ $|\boldsymbol{B}|$

[B].   $D$ $=$ $|\boldsymbol{A}| |\boldsymbol{B}|$

[C].   $D$ $=$ $(-1)^{mn}$ $|\boldsymbol{A}||\boldsymbol{B}|$

[D].   $D$ $=$ $(-1)^{m+n}$ $|\boldsymbol{A}|$ $|\boldsymbol{B}|$

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

## 问题

$\left|\begin{array}{ll} \boldsymbol{C} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right|$.

## 选项

[A].   $D$ $=$ $(-1)^{m+n}$ $|\boldsymbol{A}|$ $|\boldsymbol{B}|$

[B].   $D$ $=$ $|\boldsymbol{A}|$ $+$ $|\boldsymbol{B}|$

[C].   $D$ $=$ $|\boldsymbol{A}| |\boldsymbol{B}|$

[D].   $D$ $=$ $(-1)^{mn}$ $|\boldsymbol{A}||\boldsymbol{B}|$

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

## 问题

$\left|\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right|$.

## 选项

[A].   $D$ $=$ $|\boldsymbol{A}|$ $+$ $|\boldsymbol{B}|$

[B].   $D$ $=$ $|\boldsymbol{A}| |\boldsymbol{B}|$

[C].   $D$ $=$ $(-1)^{mn}$ $|\boldsymbol{A}||\boldsymbol{B}|$

[D].   $D$ $=$ $(-1)^{m+n}$ $|\boldsymbol{A}|$ $|\boldsymbol{B}|$

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

## 问题

$\left|\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right|$.

## 选项

[A].   $D$ $=$ $\frac{|\boldsymbol{A}|}{|\boldsymbol{B}|}$

[B].   $D$ $=$ $|\boldsymbol{A}|$ $+$ $|\boldsymbol{B}|$

[C].   $D$ $=$ $|\boldsymbol{A} \boldsymbol{B}|$

[D].   $D$ $=$ $|\boldsymbol{A}||\boldsymbol{B}|$

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

## 问题

$\left|\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right|$.

## 选项

[A].   $D$ $=$ $\frac{|\boldsymbol{A}|}{|\boldsymbol{B}|}$

[B].   $D$ $=$ $|\boldsymbol{A}|$ $+$ $|\boldsymbol{B}|$

[C].   $D$ $=$ $|\boldsymbol{A} \boldsymbol{B}|$

[D].   $D$ $=$ $|\boldsymbol{A}||\boldsymbol{B}|$

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

## 问题

$\left|\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right|$.

## 选项

[A].   $D$ $=$ $|\boldsymbol{A}|$ $+$ $|\boldsymbol{B}|$

[B].   $D$ $=$ $|\boldsymbol{A} \boldsymbol{B}|$

[C].   $D$ $=$ $|\boldsymbol{A}||\boldsymbol{B}|$

[D].   $D$ $=$ $\frac{|\boldsymbol{A}|}{|\boldsymbol{B}|}$

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

## 问题

$\left|\begin{array}{cccc} 0 & & & \lambda_{1} \\ & & \lambda_{2} & * \\ & \cdots\ & \vdots & \vdots \\ \lambda_{n} & \cdots & * & * \end{array}\right|$.

## 选项

[A].   $D$ $=$ $(-1)^{\frac{n}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[B].   $D$ $=$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[C].   $D$ $=$ $(-1)^{\frac{n(n-1)}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[D].   $D$ $=$ $(-1)^{\frac{n-1}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

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

## 问题

$\left|\begin{array}{cccc} * & \cdots & * & \lambda_{1} \\ * & \cdots & \lambda_{2} & \\ \vdots & \cdots & & \\ \lambda_{n} & & & 0 \end{array}\right|$.

## 选项

[A].   $D$ $=$ $(-1)^{\frac{n-1}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[B].   $D$ $=$ $(-1)^{\frac{n}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[C].   $D$ $=$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[D].   $D$ $=$ $(-1)^{\frac{n(n-1)}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

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

## 问题

$\left|\begin{array}{cccc} 0 & & & \lambda_{1} \\ & & \lambda_{2} & \\ & \cdots & & \\ \lambda_{n} & & & 0 \end{array}\right|$.

## 选项

[A].   $D$ $=$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[B].   $D$ $=$ $(-1)^{\frac{n(n-1)}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[C].   $D$ $=$ $(-1)^{\frac{n-1}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

[D].   $D$ $=$ $(-1)^{\frac{n}{2}}$ $\lambda_{1}$ $\lambda_{2}$ $\cdots$ $\lambda_{n}$

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

## 选项

[A].   $\begin{vmatrix} \textcolor{orange}{\circ} & \textcolor{orange}{\circ} & \textcolor{orange}{\circ}\\ \ast & \ast & \ast\\ \ast & \ast & \ast \end{vmatrix}$

[B].   $\begin{vmatrix} \textcolor{orange}{\circ} & \ast & \ast\\ \ast & \textcolor{orange}{\circ} & \ast\\ \ast & \ast & \textcolor{orange}{\circ} \end{vmatrix}$

[C].   $\begin{vmatrix} \ast & \ast & \textcolor{orange}{\circ}\\ \ast & \textcolor{orange}{\circ} & \ast\\ \textcolor{orange}{\circ} & \ast & \ast \end{vmatrix}$

[D].   $\begin{vmatrix} \textcolor{orange}{\circ} & \ast & \ast\\ \textcolor{orange}{\circ} & \ast & \ast\\ \textcolor{orange}{\circ} & \ast & \ast \end{vmatrix}$

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

## 选项

[A].   $\begin{vmatrix} \textcolor{orange}{\odot} & \ast & \ast\\ \textcolor{orange}{\odot} & \ast & \ast\\ \textcolor{orange}{\odot} & \ast & \ast \end{vmatrix}$

[B].   $\begin{vmatrix} \textcolor{orange}{\odot} & \textcolor{orange}{\odot} & \textcolor{orange}{\odot}\\ \ast & \ast & \ast\\ \ast & \ast & \ast \end{vmatrix}$

[C].   $\begin{vmatrix} \ast & \ast & \textcolor{orange}{\odot}\\ \ast & \textcolor{orange}{\odot} & \ast\\ \textcolor{orange}{\odot} & \ast & \ast \end{vmatrix}$

[D].   $\begin{vmatrix} \textcolor{orange}{\odot} & \ast & \ast\\ \ast & \textcolor{orange}{\odot} & \ast\\ \ast & \ast & \textcolor{orange}{\odot} \end{vmatrix}$

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

## 问题

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

## 选项

[A].   $D$ $=$ $\frac{1}{n}$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$

[B].   $D$ $=$ $a_{11}$ $+$ $a_{22}$ $+$ $\cdots$ $+$ $a_{n n}$

[C].   $D$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$

[D].   $D$ $=$ $\frac{1}{a_{11}}$ $\frac{1}{a_{22}}$ $\cdots$ $\frac{1}{a_{n n}}$

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

## 问题

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

## 选项

[A].   $D$ $=$ $a_{11}$ $+$ $a_{22}$ $+$ $\cdots$ $+$ $a_{n n}$

[B].   $D$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$

[C].   $D$ $=$ $\frac{1}{a_{11}}$ $\frac{1}{a_{22}}$ $\cdots$ $\frac{1}{a_{n n}}$

[D].   $D$ $=$ $\frac{1}{n}$ $a_{11}$ $a_{22}$ $\cdots$ $a_{n n}$

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