![线性代数](https://wfqqreader-1252317822.image.myqcloud.com/cover/441/32164441/b_32164441.jpg)
1.3 克莱姆(Cramer)法则
本章第一节引入了利用二阶和三阶行列式求解二元、三元线性方程组的克莱姆法则.本节将利用n阶行列式的性质,给出求解n个未知量、n个方程的线性方程组的克莱姆法则.
设n个未知量、n个方程的线性方程组为
![42692-00-023-02.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-023-02.png?sign=1738846333-49l7HMD9Rvy2Zw44o7G946e3KQoPSxtC-0-59734921a771ea02b508153bbf0a62d6)
其系数行列式
![42692-00-023-03.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-023-03.png?sign=1738846333-SxzhNlh7aPyMvz8DfQihGspFrydpjcpZ-0-420b9222f0f8d3dd4bbc4f1f95c5fa06)
下面讨论方程组(1.3.1)的求解问题.
为消去方程组(1.3.1)中的x2,x3,…,xn解出x1,用D的第一列元素的代数余子式A11,A21,…,An1分别乘以方程组(1.3.1)的第1,第2,…,第n个方程,得
![42692-00-023-04.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-023-04.png?sign=1738846333-11VHDRuA11U9RzUXQODdo0r00wtakdMK-0-2bb221acf8feb950da2d8158d4845346)
再将上面n个方程的左右两端分别相加,由式(1.2.5),有
![42692-00-023-05.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-023-05.png?sign=1738846333-fD9BFfxnDIwNuOWKYSMRG9XyNfba7kKZ-0-ccd2141c6527085d4ac4e6bda7bf8134)
即 .
同理可用D的第j(j=2,3,…,n)列元素的代数余子式A1j,A2j,…,Anj依次乘方程(1.3.1)的每一个方程,得
![42692-00-023-07.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-023-07.png?sign=1738846333-0UQVH74vUNOhn2EjbFRiQUnhabA7eqnM-0-f56961c6f747e0389c7f6a0b7a2a05f4)
记行列式
![42692-00-024-01.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-024-01.png?sign=1738846333-O34Z5zzoEHAJsE0npCDaWsFGguVayayb-0-ec03b0038c57a391610547515a347163)
Dj是把系数行列式D的第j(j=1,2,3,…,n)列换为方程组(1.3.1)的常数列b1,b2,…,bn所得到的行列式.显然,当D≠0时,方程组(1.3.1)有唯一解
![42692-00-024-02.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-024-02.png?sign=1738846333-0iDXTItg81KOiuAUsqrtK9VV28fSb03K-0-430bb8451100dab1d80eee773e804b98)
定理1.3.1(克莱姆(Cramer)法则)含有n个未知量、n个方程的线性方程组(1.3.1),当其系数行列式D≠0时,有且仅有一个解
![42692-00-024-03.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-024-03.png?sign=1738846333-aA79xJAZXaTTTWyFBq3qn6kr6oqeVIWS-0-dec3fc589b705c12d0937cb0d3bcd253)
其中,Dj是把系数行列式D的第j列换为方程组的常数列b1,b2,…,bn所得到的n阶行列式(j=1,2,3,…,n).
例1.3.1 解线性方程组
![42692-00-024-04.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-024-04.png?sign=1738846333-aEgHDAoPYLiv27uKKzYQFGvlxE5X4nhR-0-47aa73b6f0ab558a0e6e50726ef71353)
解 方程组的系数行列式
![42692-00-024-05.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-024-05.png?sign=1738846333-JYd8v89djjKMIOcEtOOUzzhaZQIQHSwu-0-4067eeb4c50196271f901656b6062c7b)
故方程组有唯一解.而
![42692-00-024-06.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-024-06.png?sign=1738846333-CKDLlKFoMliEqdDIAUN7fnFQ4YZhQHgP-0-695c5c18abf7c44a91e0d050f68ecadd)
所以方程组的解为
![42692-00-025-01.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-025-01.png?sign=1738846333-3Rz3uHEro8hLZcvoSrYLlEcqHydr0fgk-0-8e5ccf14b5d46c7d36631637d1e7748a)
如果方程组(1.3.1)的常数项全都为零,即
![42692-00-025-02.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-025-02.png?sign=1738846333-HoNHcALicYA2dSyYRA3zn6sh0joWgaJi-0-ae81c893e3086d28f02998102eaf039c)
方程组(1.3.4)称为齐次线性方程组.而方程组(1.3.1)称为非齐次线性方程组.
方程组(1.3.4)的系数行列式D≠0时,显然,x1=x2=…=xn=0一定是齐次线性方程组的解,并且是唯一的一组零解.因此,若方程组(1.3.4)具有非零解,必须D=0,即有如下定理:
定理1.3.2 含有n个未知量、n个方程的齐次线性方程组(1.3.4)若有非零解,则它的系数行列式D=0.
该定理说明系数行列式D=0是齐次线性方程组(1.3.4)有非零解的必要条件.在第四章中还将证明D=0是齐次线性方程组(1.3.4)有非零解的充分条件.
例1.3.2 问当λ为何值时,齐次线性方程组
![42692-00-025-03.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-025-03.png?sign=1738846333-aiT2dxHuTsTcbHFsJyL8UWE4DM07UZyN-0-3f58e98b0766125658ea6641f4983f0c)
只有零解?
解 方程组的系数行列式
![42692-00-025-04.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-025-04.png?sign=1738846333-BC8pRIGQikMyM0I9CazggINjb2cW1fBY-0-c31c04adbe3243f93d3337f615fa9b46)
当λ≠0,λ≠±1时,方程组只有零解.
克莱姆法则只能在D≠0时应用.D=0的情况将在第四章讨论.
习题1-3
1. 用克莱姆法则求解下列线性方程组.
![42692-00-025-05.png](https://epubservercos.yuewen.com/6F77DC/17404908804237606/epubprivate/OEBPS/Images/42692-00-025-05.png?sign=1738846333-jPLyUPGlI18QcbSHnCI87mjX8MI6DhY7-0-d404b8c998737227b5ad30663e60d74e)