科普视频
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考点
| 序号 | 题型 | 秒杀步骤 | 出题概率 | | --- | -------- | ---- | ---- | | 考点1 | 二阶行列式计算 | 4 | 100% | | 考点1 | 三阶及高阶行列式 | 4 | 100% | | 考点2 | 行列式关系 | 3 | 60% | | 考点2 | 经典行列式 | 3 | 60% | | 考点2 | 范德蒙德行列式 | 3 | 60% |
题型：二阶行列式计算
公式0：
> [!note] 公式0 > $ > \left| > \begin{array}{cc} > a_{11}& a_{12} \\ > a_{21}& a_{22} \\ > \end{array} > \right | > =a_{11}a_{22}-a_{12}a_{21} > $
例：计算行列式的值
> [!example] 例1 > $ > \left| > > \begin{array}{cc} > 2&3\\ > 1&4\\ > \end{array} > > \right| > =2\times4-1\times3 > =5 > $
> [!example] 例2 > $ > \left| > > \begin{array}{cc} > 3&3\\ > -5&4 > \end{array} > > \right| > =3\times4-(-5)\times3 > =27 > $
题型：高阶行列式计算
公式与性质
> [!note] 公式1：上三角行列式的计算公式 > $ > \left| > > \begin{array}{ccc} > > \textcolor{green}{a_{11}} & a_{12} & a_{13} \\ > \textcolor{gray}{0} & \textcolor{green}{a_{22}} & a_{23} \\ > \textcolor{gray}{0}& \textcolor{gray}{0} & \textcolor{green}{a_{33}} \\ > > \end{array} > > \right| > = > a_{11}a_{22}a_{33} > $
> [!note] 性质1 > 行列互换，行列式不变。
> [!note] 性质2 > 把行列式中某一行（列）的所有元素都乘以一个数K，等于用数K乘以行列式。
> [!example] 例 > > $ > D= > > \left| > \begin{array}{ccc} > \textcolor{magenta}{3}&\textcolor{magenta}{6}&\textcolor{magenta}{9}\\ > \textcolor{null}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \end{array} > \right| > > \xlongequal[]{性质2} > > \textcolor{red}{3} > \left| > \begin{array}{ccc} > \textcolor{red}{1}&\textcolor{red}{2}&\textcolor{red}{3}\\ > \textcolor{null}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \end{array} > \right| > $
> [!note] 性质6 > 把一行（列）的倍数加到另一行（列），行列式不变。
> [!note] 性质7 > 对换行列式中两行（列）的位置，行列式反号。
> [!example] 例 > $ > \left| > > \begin{array}{ccc} > > 1&2&3\\ > 0&1&-2\\ > 0&0&-3 > \end{array} > > \right| > = > 1\times1\times(-3) > =-3 > $
> [!hint] 归纳：解题思路 > > $ > \left| > > \begin{array}{ccc} > > a_{11} & a_{12} & a_{13} \\ > a_{21} & a_{22} & a_{23} \\ > a_{31} & a_{32} & a_{33} \\ > > \end{array} > > \right| > $ > > 设法转化为 > > $ > \left| > > \begin{array}{ccc} > > a_{11} & a_{12} & a_{13} \\ > 0 & a_{22} & a_{23} \\ > 0& 0 & a_{33} \\ > > \end{array} > > \right| > $
子题型：高阶行列式转化为上三角行列式
> [!question] 例 > 计算行列式的值： > $ > D= > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > -3&3&-7&9&-5\\ > 2&0&4&-2&1\\ > 3&-5&7&-14&6\\ > 4&-4&10&-10&2 > \end{array} > \right| > $
> [!done]- 解 > > > [!hint] 思路 > > 利用性质6、性质7构造上三角行列式 > > $ > D= > > \left| > \begin{array}{ccccc} > > \textcolor{green}{1}& > \textcolor{yellow}{-1}& > \textcolor{yellow}{2}& > \textcolor{yellow}{-3}& > \textcolor{yellow}{1}&\\ > > \textcolor{magenta}{-3}& > \textcolor{magenta}{3}& > \textcolor{magenta}{-7}& > \textcolor{magenta}{9}& > \textcolor{magenta}{-5}\\ > > 2&0&4&-2&1\\ > 3&-5&7&-14&6\\ > 4&-4&10&-10&2 > > \end{array} > \right| > > > \xlongequal[性质6,转化第2行]{ \textcolor{magenta}{r_2}+3 \textcolor{yellow}{r_1}\rightarrow \textcolor{red}{r_2}} > > \left| > \begin{array}{ccccc} > > \textcolor{green}{1}&-1&2&-3&1\\ > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{-1}& > \textcolor{red}{0}& > \textcolor{red}{-2}\\ > > 2& > 0& > 4& > -2& > 1\\ > > 3&-5&7&-14&6\\ > 4&-4&10&-10&2 > > \end{array} > \right| > $ > > $ > D= > \left| > \begin{array}{ccccc} > > \textcolor{green}{1}&-1&2&-3&1\\ > > \textcolor{yellow}{0}& > \textcolor{yellow}{0}& > \textcolor{yellow}{-1}& > \textcolor{yellow}{0}& > \textcolor{yellow}{-2}\\ > > \textcolor{magenta}{2}& > \textcolor{magenta}{0}& > \textcolor{magenta}{4}& > \textcolor{magenta}{-2}& > \textcolor{magenta}{1}\\ > > 3&-5&7&-14&6\\ > 4&-4&10&-10&2 > > \end{array} > \right| > > \xlongequal[性质6,转化第3行]{\textcolor{magenta}{r_3}-2\textcolor{yellow}{r_1}\rightarrow \textcolor{red}{r_3}} > > \left| > \begin{array}{ccccc} > \textcolor{green}{1}&-1&2&-3&1\\ > 0&0&-1&0&-2\\ > > \textcolor{red}{0}& > \textcolor{red}{2}& > \textcolor{red}{0}& > \textcolor{red}{4}& > \textcolor{red}{-1}\\ > > \textcolor{null}{3}& > \textcolor{null}{-5}& > \textcolor{null}{7}& > \textcolor{null}{-14}& > \textcolor{null}{6}\\ > > \textcolor{null}{4}& > \textcolor{null}{-4}& > \textcolor{null}{10}& > \textcolor{null}{-10}& > \textcolor{null}{2} > > \end{array} > \right| > $ > > $ > D= > \left| > \begin{array}{ccccc} > \textcolor{green}{1}& > \textcolor{yellow}{-1}& > \textcolor{yellow}{2}& > \textcolor{yellow}{-3}& > \textcolor{yellow}{1}\\ > > > 0&0&-1&0&-2\\ > > \textcolor{null}{0}& > \textcolor{null}{2}& > \textcolor{null}{0}& > \textcolor{null}{4}& > \textcolor{null}{-1}\\ > > \textcolor{magenta}{3}& > \textcolor{magenta}{-5}& > \textcolor{magenta}{7}& > \textcolor{magenta}{-14}& > \textcolor{magenta}{6}\\ > > \textcolor{magenta}{4}& > \textcolor{magenta}{-4}& > \textcolor{magenta}{10}& > \textcolor{magenta}{-10}& > \textcolor{magenta}{2} > > \end{array} > \right| > \xlongequal[性质6,转化第4、5行]{\textcolor{magenta}{r_4}-3\textcolor{yellow}{r_1}\rightarrow \textcolor{red}{r_4},\textcolor{magenta}{r_5}-4\textcolor{yellow}{r_1}\rightarrow \textcolor{red}{r_5}} > > \left| > \begin{array}{ccccc} > \textcolor{green}{1}&-1&2&-3&1\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{null}{-1}& > \textcolor{null}{0}& > \textcolor{null}{-2}\\ > > \textcolor{null}{0}& > \textcolor{null}{2}& > \textcolor{null}{0}& > \textcolor{null}{4}& > \textcolor{null}{-1}\\ > > \textcolor{red}{0}& > \textcolor{red}{-2}& > \textcolor{red}{1}& > \textcolor{red}{-5}& > \textcolor{red}{3}\\ > > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{2}& > \textcolor{red}{2}& > \textcolor{red}{-2} > > \end{array} > \right| > $ > > $ > D= > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > > \textcolor{magenta}{0}& > \textcolor{magenta}{0}& > \textcolor{magenta}{-1}& > \textcolor{magenta}{0}& > \textcolor{magenta}{-2}\\ > > \textcolor{magenta}{0}& > \textcolor{magenta}{2}& > \textcolor{magenta}{0}& > \textcolor{magenta}{4}& > \textcolor{magenta}{-1}\\ > > \textcolor{null}{0}& > \textcolor{null}{-2}& > \textcolor{null}{1}& > \textcolor{null}{-5}& > \textcolor{null}{3}\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{null}{2}& > \textcolor{null}{2}& > \textcolor{null}{-2} > > \end{array} > \right| > \xlongequal[性质7，将2、3行互换]{\textcolor{magenta}{r_2}\leftrightarrow \textcolor{magenta}{r_3}} > \textcolor{red}{-} > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > > \textcolor{red}{0}& > \textcolor{red}{2}& > \textcolor{red}{0}& > \textcolor{red}{4}& > \textcolor{red}{-1}\\ > > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{-1}& > \textcolor{red}{0}& > \textcolor{red}{-2}\\ > > \textcolor{null}{0}& > \textcolor{null}{-2}& > \textcolor{null}{1}& > \textcolor{null}{-5}& > \textcolor{null}{3}\\ > > 0&0&2&2&-2 > \end{array} > \right| > $ > > $ > D= > - > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > > \textcolor{yellow}{0}& > \textcolor{green}{2}& > \textcolor{yellow}{0}& > \textcolor{yellow}{4}& > \textcolor{yellow}{-1}\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{null}{-1}& > \textcolor{null}{0}& > \textcolor{null}{-2}\\ > > \textcolor{magenta}{0}& > \textcolor{magenta}{-2}& > \textcolor{magenta}{1}& > \textcolor{magenta}{-5}& > \textcolor{magenta}{3}\\ > > 0&0&2&2&-2 > \end{array} > \right| > \xlongequal[性质6,转化第4行]{\textcolor{magenta}{r_4}+\textcolor{yellow}{r_2}\rightarrow \textcolor{red}{r_4}} > - > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > 0&\textcolor{green}{2}&0&4&-1\\ > 0&0&-1&0&-2\\ > > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{1}& > \textcolor{red}{-1}& > \textcolor{red}{2}\\ > > 0&0&2&2&-2 > \end{array} > \right| > $ > > > $ > D= > - > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > 0&\textcolor{null}{2}&0&4&-1\\ > > \textcolor{yellow}{0}& > \textcolor{yellow}{0}& > \textcolor{green}{-1}& > \textcolor{yellow}{0}& > \textcolor{yellow}{-2}&\\ > > \textcolor{magenta}{0}& > \textcolor{magenta}{0}& > \textcolor{magenta}{1}& > \textcolor{magenta}{-1}& > \textcolor{magenta}{2}\\ > > \textcolor{magenta}{0}& > \textcolor{magenta}{0}& > \textcolor{magenta}{2}& > \textcolor{magenta}{2}& > \textcolor{magenta}{-2}\\ > > \end{array} > \right| > > \xlongequal[性质6,转化第4、5行]{ > \textcolor{magenta}{r_4}+\textcolor{yellow}{r_3}\rightarrow \textcolor{red}{r_4}, > \textcolor{magenta}{r_5}+2\textcolor{yellow}{r_3}\rightarrow \textcolor{red}{r_5} > } > > - > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > > \textcolor{null}{0}& > \textcolor{null}{2}& > \textcolor{null}{0}& > \textcolor{null}{4}& > \textcolor{null}{-1}\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{green}{-1}& > \textcolor{null}{0}& > \textcolor{null}{-2}\\ > > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{-1}& > \textcolor{red}{0}\\ > > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{2}& > \textcolor{red}{-6} > \end{array} > \right| > $ > > $ > D= > - > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > > \textcolor{null}{0}& > \textcolor{null}{2}& > \textcolor{null}{0}& > \textcolor{null}{4}& > \textcolor{null}{-1}\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{null}{-1}& > \textcolor{null}{0}& > \textcolor{null}{-2}\\ > > \textcolor{yellow}{0}& > \textcolor{yellow}{0}& > \textcolor{yellow}{0}& > \textcolor{green}{-1}& > \textcolor{yellow}{0}\\ > > \textcolor{magenta}{0}& > \textcolor{magenta}{0}& > \textcolor{magenta}{0}& > \textcolor{magenta}{2}& > \textcolor{magenta}{-6} > \end{array} > \right| > \xlongequal[性质6,转化第5行]{\textcolor{magenta}{r_5}+2\textcolor{yellow}{r_4}\rightarrow\textcolor{red}{r_5}} > > - > \left| > \begin{array}{ccccc} > 1&-1&2&-3&1\\ > > \textcolor{null}{0}& > \textcolor{null}{2}& > \textcolor{null}{0}& > \textcolor{null}{4}& > \textcolor{null}{-1}\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{null}{-1}& > \textcolor{null}{0}& > \textcolor{null}{-2}\\ > > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{null}{0}& > \textcolor{green}{-1}& > \textcolor{null}{0}\\ > > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{0}& > \textcolor{red}{-6} > \end{array} > \right| > $ > > > [!hint] 思路 > > 利用公式1，直接计算值 > > $ > D= > - > \left| > \begin{array}{ccccc} > \textcolor{magenta}{1}&\textcolor{null}{-1}&\textcolor{null}{2}&\textcolor{null}{-3}&\textcolor{null}{1}\\ > \textcolor{gray}{0}&\textcolor{magenta}{2}&\textcolor{null}{0}&\textcolor{null}{4}&\textcolor{null}{-1}\\ > \textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{magenta}{-1}&\textcolor{null}{0}&\textcolor{null}{-2}\\ > \textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{magenta}{-1}&\textcolor{null}{0}\\ > \textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{magenta}{-6} > \end{array} > \right| > \xlongequal[]{公式1} > -[\textcolor{magenta}{1}\times\textcolor{magenta}{2}\times(\textcolor{magenta}{-1})\times(\textcolor{magenta}{-1})\times(\textcolor{magenta}{-6})] > =\textcolor{red}{12} > $
子题型：三阶行列式转化为上三角行列式
> [!question] 题干 > 计算行列式： > $ > \left| > \begin{array}{ccc} > \textcolor{green}{1}&\textcolor{yellow}{2}&\textcolor{yellow}{3}\\ > \textcolor{magenta}{2}&\textcolor{magenta}{3}&\textcolor{magenta}{1}\\ > \textcolor{magenta}{3}&\textcolor{magenta}{1}&\textcolor{magenta}{2}\\ > \end{array} > \right| > $
> [!done]- 解 > > [!hint] 思路 > > 利用性质6、性质7构造上三角行列式 > > $ > D= > > \left| > > \begin{array}{ccc} > \textcolor{green}{1}&\textcolor{yellow}{2}&\textcolor{yellow}{3}\\ > \textcolor{magenta}{2}&\textcolor{magenta}{3}&\textcolor{magenta}{1}\\ > \textcolor{magenta}{3}&\textcolor{magenta}{1}&\textcolor{magenta}{2}\\ > \end{array} > > \right| > > \xlongequal[公式1，转化第2、3行]{ > \textcolor{magenta}{r_2}-2\textcolor{yellow}{r_1}\rightarrow\textcolor{red}{r_2}, > \textcolor{magenta}{r_3}-3\textcolor{yellow}{r_1}\rightarrow\textcolor{red}{r_3} > } > > \left| > > \begin{array}{ccc} > \textcolor{green}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{red}{0}&\textcolor{red}{-1}&\textcolor{red}{-5}\\ > \textcolor{red}{0}&\textcolor{red}{-5}&\textcolor{red}{-7}\\ > \end{array} > > \right| > $ > > $ > D= > > > > > > \left| > > \begin{array}{ccc} > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{yellow}{0}&\textcolor{green}{-1}&\textcolor{yellow}{-5}\\ > \textcolor{magenta}{0}&\textcolor{magenta}{-5}&\textcolor{magenta}{-7}\\ > \end{array} > > \right| > > \xlongequal[公式1，转化第2、3行]{ > \textcolor{magenta}{r_3}-3\textcolor{yellow}{r_1}\rightarrow\textcolor{red}{r_3} > } > > \left| > > \begin{array}{ccc} > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{null}{0}&\textcolor{green}{-1}&\textcolor{null}{-5}\\ > \textcolor{red}{0}&\textcolor{red}{0}&\textcolor{red}{18}\\ > \end{array} > > \right| > $ > > > [!hint] 思路 > > 利用公式1，直接计算值 > > $ > D= > > \left| > \begin{array}{ccc} > \textcolor{magenta}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{gray}{0}&\textcolor{magenta}{-1}&\textcolor{null}{-5}\\ > \textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{magenta}{18}\\ > \end{array} > \right| > > \xlongequal[]{公式1} > > \textcolor{magenta}{1}*(\textcolor{magenta}{-1})*\textcolor{magenta}{18}= > > \textcolor{red}{-18} > $
技巧：利用性质7，构造对角线参照项的值为1
> [!question] 例题 > 计算行列式 > > $ > D= > > \left| > \begin{array}{ccc} > \textcolor{null}{3}&\textcolor{null}{1}&\textcolor{null}{2}\\ > \textcolor{null}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3} > \end{array} > \right| > $
> [!done]- 解 > > [!hint] 思路 > > 利用性质7，构造对角线参照项的值为1 > > $ > D= > > \left| > \begin{array}{ccc} > \textcolor{magenta}{3}&\textcolor{magenta}{1}&\textcolor{magenta}{2}\\ > \textcolor{null}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{magenta}{1}&\textcolor{magenta}{2}&\textcolor{magenta}{3} > \end{array} > \right| > > \xlongequal[性质7，兑换第1、3行]{\textcolor{magenta}{r_1}\leftrightarrow \textcolor{magenta}{r_3}} > \textcolor{red}{-} > \left| > \begin{array}{ccc} > \textcolor{red}{1}&\textcolor{red}{2}&\textcolor{red}{3}\\ > \textcolor{null}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{red}{3}&\textcolor{red}{1}&\textcolor{red}{2}\\ > \end{array} > \right| > $ > > > [!hint] 思路 > > 利用性质6、性质7构造上三角行列式 > > $ > D= > \textcolor{null}{-} > \left| > \begin{array}{ccc} > \textcolor{green}{1}&\textcolor{yellow}{2}&\textcolor{yellow}{3}\\ > \textcolor{magenta}{2}&\textcolor{magenta}{3}&\textcolor{magenta}{1}\\ > \textcolor{magenta}{3}&\textcolor{magenta}{1}&\textcolor{magenta}{2}\\ > \end{array} > \right| > > \xlongequal[性质6，转化第2、3行]{ > \textcolor{magenta}{r_3}-3\textcolor{yellow}{r_1}\rightarrow \textcolor{red}{r_3}, > \textcolor{magenta}{r_2}-2\textcolor{yellow}{r_1}\rightarrow \textcolor{red}{r_2} > } > > \textcolor{null}{-} > \left| > \begin{array}{ccc} > \textcolor{green}{1}&\textcolor{yellow}{2}&\textcolor{yellow}{3}\\ > \textcolor{red}{0}&\textcolor{red}{-1}&\textcolor{red}{-5}\\ > \textcolor{red}{0}&\textcolor{red}{-5}&\textcolor{red}{-7}\\ > \end{array} > \right| > $ > > $ > D= > \textcolor{null}{-} > \left| > \begin{array}{ccc} > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{yellow}{0}&\textcolor{green}{-1}&\textcolor{yellow}{-5}\\ > \textcolor{magenta}{0}&\textcolor{magenta}{-5}&\textcolor{magenta}{-7}\\ > \end{array} > \right| > \xlongequal[性质6，转化第3行]{\textcolor{magenta}{r_3}-5\textcolor{yellow}{r_2}\rightarrow\textcolor{red}{r_3} > > } > \textcolor{null}{-} > \left| > \begin{array}{ccc} > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{yellow}{0}&\textcolor{green}{-1}&\textcolor{yellow}{-5}\\ > \textcolor{red}{0}&\textcolor{red}{0}&\textcolor{red}{18}\\ > \end{array} > \right| > $ > > $ > D= > - > \left| > \begin{array}{ccc} > \textcolor{magenta}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{gray}{0}&\textcolor{magenta}{-1}&\textcolor{null}{-5}\\ > \textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{magenta}{18}\\ > \end{array} > \right| > \xlongequal[]{公式1} > - > [\textcolor{magenta}{1}*(\textcolor{magenta}{-1})*\textcolor{magenta}{18}] > =\textcolor{red}{18} > $ >
公式2
> [!note] 公式2 > $ > D= > > \left| > \begin{array}{cccc} > \textcolor{null}{x}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right| > = > \textcolor{null}{[x+(n-1)a]}\textcolor{null}{(x-a)^{n-1}} > $
> [!done]- 证明 > > ##### 第1大步骤：目的：把所有行都加到第一行上，得到固定值 > > > $ > D= > > \left| > \begin{array}{cccc} > \textcolor{green}{x}&\textcolor{magenta}{a}&\textcolor{magenta}{...}&\textcolor{magenta}{a}\\ > \textcolor{yellow}{a}&\textcolor{green}{x}&\textcolor{yellow}{...}&\textcolor{yellow}{a}\\ > \textcolor{yellow}{...}&\textcolor{yellow}{...}&\textcolor{yellow}{...}&\textcolor{yellow}{...}\\ > \textcolor{yellow}{a}&\textcolor{yellow}{a}&\textcolor{yellow}{...}&\textcolor{green}{x}\\ > \end{array} > \right| > > \xlongequal[全部加到第1行]{性质6，变换第1行} > > \left| > \begin{array}{cccc} > \textcolor{red}{x+(n-1)a}&\textcolor{red}{x+(n-1)a}&\textcolor{red}{...}&\textcolor{red}{x+(n-1)a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right| > $ > > > > ###### 归纳过程 > > > - 正向归纳 > > 第1行+第2行 > $ > D= > \left| > \begin{array}{cccc} > \textcolor{magenta}{x}&\textcolor{magenta}{a}&\textcolor{magenta}{...}&\textcolor{magenta}{a}\\ > \textcolor{yellow}{a}&\textcolor{green}{x}&\textcolor{yellow}{...}&\textcolor{yellow}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right| > \xlongequal[]{\textcolor{magenta}{r_1}+\textcolor{yellow}{r_2}\rightarrow \textcolor{red}{r_1}} > \left| > \begin{array}{cccc} > \textcolor{red}{x+a}&\textcolor{red}{a+x}&\textcolor{red}{...}&\textcolor{red}{a+a}\\ > \textcolor{null}{a}&\textcolor{green}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right|\\ > $ > > 第1行+第3行 > > $ > D= > \left| > \begin{array}{cccc} > \textcolor{magenta}{x+a}&\textcolor{magenta}{a+x}&\textcolor{magenta}{a+a}&\textcolor{magenta}{...}&\textcolor{magenta}{a+a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{yellow}{a}&\textcolor{yellow}{a}&\textcolor{green}{x}&\textcolor{yellow}{...}&\textcolor{yellow}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right| > \xlongequal[]{\textcolor{magenta}{r_1}+\textcolor{yellow}{r_3}\rightarrow \textcolor{red}{r_1}} > \left| > \begin{array}{cccc} > \textcolor{red}{x+a+a}&\textcolor{red}{a+x+a}&\textcolor{red}{a+a+x}&\textcolor{red}{...}&\textcolor{red}{a+a+a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{green}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right|\\ > > ...\\ > $ > > ... > > - 逆向归纳 > > 第1行+第n行 > > > $ > D= > \left| > \begin{array}{cccc} > \textcolor{magenta}{x+(n-2)a}&\textcolor{magenta}{x+(n-2)a}&\textcolor{magenta}{x+(n-2)a}&\textcolor{magenta}{...}&\textcolor{magenta}{x+(n-2)a}&\textcolor{magenta}{(n-1)a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{a}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}&\textcolor{null}{a}\\ > > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}&\textcolor{null}{a}\\ > \textcolor{yellow}{a}&\textcolor{yellow}{a}&\textcolor{yellow}{a}&\textcolor{yellow}{...}&\textcolor{yellow}{a}&\textcolor{green}{x}\\ > \end{array} > \right|\\ > \xlongequal[]{\textcolor{magenta}{r_1}+\textcolor{yellow}{r_{n}}\rightarrow \textcolor{red}{r_1}} > \left| > \begin{array}{cccc} > \textcolor{red}{x+(n-1)a}&\textcolor{red}{x+(n-1)a}&\textcolor{red}{x+(n-1)a}&\textcolor{red}{...}&\textcolor{red}{x+(n-1)a}&\textcolor{red}{x+(n-1)a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{a}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}&\textcolor{null}{a}\\ > > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{a}&\textcolor{green}{x}\\ > \end{array} > \right|\\ > \xlongequal[]{整理结果} > \left| > \begin{array}{cccc} > \textcolor{red}{x+(n-1)a}&\textcolor{red}{x+(n-1)a}&\textcolor{red}{...}&\textcolor{red}{x+(n-1)a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right| > $ > > ##### 第2大步骤：目的：提取公因式 > > $ > D= > > \left| > \begin{array}{cccc} > \textcolor{magenta}{x+(n-1)a}&\textcolor{magenta}{x+(n-1)a}&\textcolor{magenta}{...}&\textcolor{magenta}{x+(n-1)a}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right|\\ > \xlongequal[]{性质2} > \textcolor{red}{[x+(n-1)a]} > \left| > \begin{array}{cccc} > \textcolor{red}{1}&\textcolor{red}{1}&\textcolor{red}{...}&\textcolor{red}{1}\\ > \textcolor{null}{a}&\textcolor{null}{x}&\textcolor{null}{...}&\textcolor{null}{a}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{...}&\textcolor{null}{x}\\ > \end{array} > \right|\\ > $ > > ##### 第3大步骤：2~n行每行都减去第1行的a倍 > > 归纳 > > $ > D= > \textcolor{null}{[x+(n-1)a]} > \left| > \begin{array}{cccc} > \textcolor{green}{1}&\textcolor{yellow}{1}&\textcolor{yellow}{...}&\textcolor{yellow}{1}\\ > \textcolor{magenta}{a}&\textcolor{magenta}{x}&\textcolor{magenta}{...}&\textcolor{magenta}{a}\\ > \textcolor{magenta}{...}&\textcolor{magenta}{...}&\textcolor{magenta}{...}&\textcolor{magenta}{...}\\ > \textcolor{magenta}{a}&\textcolor{magenta}{a}&\textcolor{magenta}{...}&\textcolor{magenta}{x}\\ > \end{array} > \right|\\ > \xlongequal[性质6]{\textcolor{magenta}{r_{2到n每一行}}-a r_1\rightarrow \textcolor{red}{r_{{2到n每一行}}}}\textcolor{null}{ > [x+(n-1)a]} > \left| > \begin{array}{cccc} > \textcolor{null}{1}&\textcolor{null}{1}&\textcolor{null}{...}&\textcolor{null}{1}\\ > \textcolor{red}{0}&\textcolor{red}{x-a}&\textcolor{red}{...}&\textcolor{red}{0}\\ > \textcolor{red}{...}&\textcolor{red}{...}&\textcolor{red}{...}&\textcolor{red}{...}\\ > \textcolor{red}{0}&\textcolor{red}{0}&\textcolor{red}{...}&\textcolor{red}{x-a}\\ > \end{array} > \right|\\ > $ > > 利用公式1 > > $ > D= > [x+(n-1)a] > \left| > \begin{array}{cccc} > \textcolor{green}{1}&\textcolor{null}{1}&\textcolor{null}{...}&\textcolor{null}{1}\\ > \textcolor{gray}{0}&\textcolor{green}{x-a}&\textcolor{null}{...}&\textcolor{null}{0}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{gray}{0}&\textcolor{gray}{0}&\textcolor{gray}{...}&\textcolor{green}{x-a}\\ > \end{array} > \right|\\ > \xlongequal[]{公式1} > \textcolor{null}{[x+(n-1)a]}\textcolor{red}{(x-a)^{n-1}} > $ >
公式2练习题
> [!question] 请计算 > $ > D= > \left| > \begin{array}{cccc} > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{b}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{b}\\ > \textcolor{null}{b}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{b}&\textcolor{null}{a}&\textcolor{null}{a}\\ > \end{array} > \right| > $
> [!hint]- 思路 > 列变换
> [!done]- 解 > $ > D= > \left| > \begin{array}{cccc} > \textcolor{magenta}{a}&\textcolor{null}{a}&\textcolor{magenta}{b}&\textcolor{null}{a}\\ > \textcolor{magenta}{a}&\textcolor{null}{a}&\textcolor{magenta}{a}&\textcolor{null}{b}\\ > \textcolor{magenta}{b}&\textcolor{null}{a}&\textcolor{magenta}{a}&\textcolor{null}{a}\\ > \textcolor{magenta}{a}&\textcolor{null}{b}&\textcolor{magenta}{a}&\textcolor{null}{a}\\ > \end{array} > \right| > \xlongequal[性质7]{ > \textcolor{magenta}{c_1} \leftrightarrow \textcolor{magenta}{c_3} > } > \textcolor{red}{-} > \left| > \begin{array}{cccc} > \textcolor{red}{b}&\textcolor{null}{a}&\textcolor{red}{a}&\textcolor{null}{a}\\ > \textcolor{red}{a}&\textcolor{null}{a}&\textcolor{red}{a}&\textcolor{null}{b}\\ > \textcolor{red}{a}&\textcolor{null}{a}&\textcolor{red}{b}&\textcolor{null}{a}\\ > \textcolor{red}{a}&\textcolor{null}{b}&\textcolor{red}{a}&\textcolor{null}{a}\\ > \end{array} > \right| > $ > > $ > D= > \textcolor{magenta}{-} > \left| > \begin{array}{cccc} > \textcolor{null}{b}&\textcolor{magenta}{a}&\textcolor{null}{a}&\textcolor{magenta}{a}\\ > \textcolor{null}{a}&\textcolor{magenta}{a}&\textcolor{null}{a}&\textcolor{magenta}{b}\\ > \textcolor{null}{a}&\textcolor{magenta}{a}&\textcolor{null}{b}&\textcolor{magenta}{a}\\ > \textcolor{null}{a}&\textcolor{magenta}{b}&\textcolor{null}{a}&\textcolor{magenta}{a}\\ > \end{array} > \right| > \xlongequal[性质7]{ > \textcolor{magenta}{c_2} \leftrightarrow \textcolor{magenta}{c_4} > } > \left| > \begin{array}{cccc} > \textcolor{null}{b}&\textcolor{red}{a}&\textcolor{null}{a}&\textcolor{red}{a}\\ > \textcolor{null}{a}&\textcolor{red}{b}&\textcolor{null}{a}&\textcolor{red}{a}\\ > \textcolor{null}{a}&\textcolor{red}{a}&\textcolor{null}{b}&\textcolor{red}{a}\\ > \textcolor{null}{a}&\textcolor{red}{a}&\textcolor{null}{a}&\textcolor{red}{b}\\ > \end{array} > \right| > $ > > $ > D= > \left| > \begin{array}{cccc} > \textcolor{green}{b}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{green}{b}&\textcolor{null}{a}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{green}{b}&\textcolor{null}{a}\\ > \textcolor{null}{a}&\textcolor{null}{a}&\textcolor{null}{a}&\textcolor{green}{b}\\ > \end{array} > \right| > \xlongequal[公式2]{ > > } > > [b+(4-1)a](b-a)^{4-1} > =(b+3a)(b-a)^3 > $
范德蒙德行列式
> [!note] 范德蒙德行列式 > $ > D_n= > \left| > \begin{array}{cccc} > \textcolor{null}{1}&\textcolor{null}{1}&\textcolor{null}{...}&\textcolor{null}{1}\\ > \textcolor{null}{x_1}&\textcolor{null}{x_2}&\textcolor{null}{...}&\textcolor{null}{x_n}\\ > \textcolor{null}{x_1^2}&\textcolor{null}{x_2^2}&\textcolor{null}{...}&\textcolor{null}{x_n^2}\\ > \textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}&\textcolor{null}{...}\\ > \textcolor{null}{x_1^{n-1}}&\textcolor{null}{x_2^{n-1}}&\textcolor{null}{...}&\textcolor{null}{x_n^{n-1}}\\ > \end{array} > \right| > \\= > \textcolor{green}{[(x_2-x_1)(x_3-x_1)...(x_n-x_1)]}\textcolor{yellow}{[(x_3-x_2)(x_4-x_2)...(x_n-x_2)]}...\textcolor{green}{(x_n-x_{n-1})} > $
题型：行列式之间的关系
> [!question] 例1 > > $ > D= > \left| > \begin{array}{ccc} > \textcolor{null}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{null}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{null}{3}&\textcolor{null}{1}&\textcolor{null}{2}\\ > \end{array} > \right| > =2,\\ > $ > $ > D_2= > \left| > \begin{array}{ccc} > \textcolor{null}{2}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{null}{4}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{null}{6}&\textcolor{null}{1}&\textcolor{null}{2}\\ > \end{array} > \right| > =? > $
> [!done]- 解 > > $ > D= > \left| > \begin{array}{ccc} > \textcolor{yellow}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{yellow}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{yellow}{3}&\textcolor{null}{1}&\textcolor{null}{2}\\ > \end{array} > \right| > =2,\\ > $ > $ > D_2= > \left| > \begin{array}{ccc} > \textcolor{yellow}{2}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{yellow}{4}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{yellow}{6}&\textcolor{null}{1}&\textcolor{null}{2}\\ > \end{array} > \right| > =2 > \left| > \begin{array}{ccc} > \textcolor{red}{1}&\textcolor{null}{2}&\textcolor{null}{3}\\ > \textcolor{red}{2}&\textcolor{null}{3}&\textcolor{null}{1}\\ > \textcolor{red}{3}&\textcolor{null}{1}&\textcolor{null}{2}\\ > \end{array} > \right| > =2D > $
> [!note] 性质5 > 如果行列式中两行（列）成比例，那么行列式为零。
> [!example] 例 > $ > D_3= > \left| > \begin{array}{ccc} > \textcolor{yellow}{1}&\textcolor{yellow}{2}&\textcolor{null}{3}\\ > \textcolor{yellow}{2}&\textcolor{yellow}{4}&\textcolor{null}{1}\\ > \textcolor{yellow}{3}&\textcolor{yellow}{6}&\textcolor{null}{2}\\ > \end{array} > \right| > =0 > $ > > > [!done]- 例子的证明 > > 例： > > $ > > D_3= > > \left| > > \begin{array}{ccc} > > \textcolor{null}{1}&\textcolor{magenta}{2}&\textcolor{null}{3}\\ > > \textcolor{null}{2}&\textcolor{magenta}{4}&\textcolor{null}{1}\\ > > \textcolor{null}{3}&\textcolor{magenta}{6}&\textcolor{null}{2}\\ > > \end{array} > > \right| > > \xlongequal[]{c_2-2c_1} > > \left| > > \begin{array}{ccc} > > \textcolor{null}{1}&\textcolor{red}{0}&\textcolor{null}{3}\\ > > \textcolor{null}{2}&\textcolor{red}{0}&\textcolor{null}{1}\\ > > \textcolor{null}{3}&\textcolor{red}{0}&\textcolor{null}{2}\\ > > \end{array} > > \right| > > $ > > > > $ > > \left| > > \begin{array}{ccc} > > \textcolor{null}{1}&\textcolor{yellow}{0}&\textcolor{null}{3}\\ > > \textcolor{null}{2}&\textcolor{green}{0}&\textcolor{null}{1}\\ > > \textcolor{null}{3}&\textcolor{yellow}{0}&\textcolor{null}{2}\\ > > \end{array} > > \right| > > \xlongequal[]{转化为上三角行列式，公式1} > > \textcolor{red}{0} > > $
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