本科生毕业论文
题 目： 行列式的计算方法及其在线性方程组中的应用姓 名：
学 号：
系 别：
年 级:
专 业：
摘 要
《高等代数》是数学专业学生的一门必修基础课程。行列式的计算是高等代数中的重点、难点，特别是n 阶行列式的计算，学生在学习过程中，普遍存在很多困难，难于掌握。计算n 阶行列式的方法很多，但具体到一个题，要针对其特征，选取适当的方法求解。当看到一个貌似非常复杂的n 阶行列式时，
仔细观察，
会发现其实它们的元素在行或列的排列方式上都有某些规律。掌握住这些规律，选择合适的计算方法，能使我们在极短的时间内达到事半功倍的效果！本文首先介绍n 阶行列式的定义、性质，再归纳总结行列式的各种计算方法、技巧及其在线性方程组中的初步应用。行列式是线性方程组理论的一个组成部分，是中学数学有关内容的提高和推广。它不仅是解线性方程组的重要工具，而且在其它一些学科分支中也有广泛的应用。
关键词：n 阶行列式 计算 方法 归纳
ABST RACT
线性方程组
Algebra is a courses of mathematics specialized compulsory of the basic mathematic. The determinant's calculation is the most difficulty in higher algebra, especially, the n order determinant's calculation , alway is student's difficulty in the learning process, so ,it is difficult to master for ours . There are a lot of calculations of
n order determinant in method , but when we say a problem of the calculation of n order determinant , according to its characteristics, selecting the appropriate method to solving is a very good idea. When you see a seemingly so complex n order determinant, we should observe them carefully,and we will find that their elements are arranged in rows or columns have some regularity. Grasping of these laws, finding a appropriate calculation method,it can help us to achieve a multiplier effect in a very short time! This paper mainly introduces the definition of n order determinant, nature, and calculation methods, the skills of calculation of n order determinant and application in linear equation group. Determinant is an important theory in linear equations and it is an indispensable part of linear equations , determinant is also the middle school mathematics' content raise and promotion. It is not only the solution of linear equations of the important tool, but also in some other branch has a wide range of applications.
Key words: n order determinant calculation method induce linear equations
目 录
引言„„„„„„„„„„„„„„„„„„„„„ 1 1 n 阶行列式的定义„„„„„„„„„„„„„„„ 3
2 n 阶行列式的性质„„„„„„„„„„„„„„„ 3 3 计算n 阶行列式的具体方法与技巧„„„„„„„„ 4
3.1 利用行列式定义直接计算„„„„„„„„„„ 4
3.2 利用行列式的性质计算„„„„„„„„„„„ 5
3.3 化为三角形行列式„„„„„„„„„„„„„ 6
3.4 降阶法„„„„„„„„„„„„„„„„„„ 7
3.5 逆推公式法„„„„„„„„„„„„„„„„ 8
3.6 利用范德蒙德行列式„„„„„„„„„„„„ 9
3.7 加边法（升阶法）„„„„„„„„„„„„„ 9
3.8 数学归纳法„„„„„„„„„„„„„„„„ 10
3.9 拆开法„„„„„„„„„„„„„„„„„„ 11 4 行列式在线性方程组中的初步应用„„„„„„„ 11
4.1 克拉默（Gramer ）法则„„„„„„„„„„„ 12
4.2 克拉默（Gramer ）法则的应用„„„„„„„„ 12
4.2.1 用克拉默（Gramer ）法则解线性方程组„„„ 13
4.2.2 克拉默法则及其推论在几何上的应用„„„„ 14 结论„„„„„„„„„„„„„„„„„„„„„ 16 参考文献„„„„„„„„„„„„„„„„„„„ 17 致谢„„„„„„„„„„„„„„„„„„„„„ 18
引 言
解方程是代数中一个基本问题，特别是在中学中所学的代数中，解方程占有重要的地位. 因此这个问题是读者所熟悉的. 比如说，如果我们知道了一段导线的电阻r ，它的两端的电位差v ，那么通过这段导线的电流强度i ，就可以有关系式
ir =v
求出来. 这就是所谓解一元一次方程的问题. 在中学所学代数中，我们解过一元、二元、三元以至四元一次方程组.
线性方程组的理论在数学中是基本的也是重要的内容.
对于二元线性方程组
⎧a x +a x =b ⎨1111221，
⎩a 21x 2+a 22x 2=b 2
当a 11a 22-a 12a 21≠0时，次方程组有惟一解，即 x 1=b 1a 22-a 12b 2a b -a b ， x 2=112121. a 11a 22-a 12a 21a 11a 22-a 12a 21
我们称a 11a 22-a 12a 21为二级行列式，用符号表示为
a 11a 22-a 12a 21=a 11a 12
a 21a 22
于是上述解可以用二级行列式叙述为：当二级行列式
时，该方程组有惟一解，即 a 11a 12a 21a 22≠0
b 1a 12
x 1=a 11b 1b 2a 22a b , x 2=212. a 11a 12a 11a 12
a 21a 22a 21a 22
对于三元线性方程组有相仿的结论. 设有三元线性方程组
⎧a 11x 1+a 12x 2+a 13x 3=b 1, ⎪ ⎨a 21x 1+a 22x 2+a 23x 3=b 2,
⎪a x +a x +a x =b . 3333⎩311322
称代数式a 11a 22a 33+a 12a 23a 31+a 13a 21a 32-a 11a 23a 32-a 12a 21a 33-a 13a 22a 31为三级行列式，用符号表示为：
a 11a 12a 13
a 11a 22a 33+a 12a 23a 31+a 13a 21a 32-a 11a 23a 32-a 12a 21a 33-a 13a 22a 31=a 21a 22a 23.
a 31a 32a 33
我们有：当三级行列式
a 11a 12a 13
d =a 21a 22a 23≠0
a 31a 32a 33
时，上述三元线性方程组有惟一解，解为 x 1=其中 d d 1d ，x 2=2，x 3=3 d d d
a 11b 1a 13
a 31b 3a 33a 11a 12b 1a 31a 32b 3b 1a 12a 13b 3a 32a 33 d 1=b 2a 22a 23 ，d 2=a 21b 2a 23，d 3=a 21a 22b 2
在本论文中我们将把这个结果推广到n 元线性方程组
⎧a 11x 1+a 12x 2+ +a 1n x n =b 1⎪a x +a x + +a x =b ⎪2112222n n 2 ⎨ ⎪⎪⎩a n 1x 1+a n 2x 2+ +a nn x n =b n
的情形. 为此，我们首先要给出n 阶行列式的定义并讨论它的性质，这就是本论文的主要内容.
本科生毕业论文
题 目： 行列式的计算方法及其在线性方程组中的应用姓 名：
学 号：
系 别：
年 级:
专 业：
摘 要
《高等代数》是数学专业学生的一门必修基础课程。行列式的计算是高等代数中的重点、难点，特别是n 阶行列式的计算，学生在学习过程中，普遍存在很多困难，难于掌握。计算n 阶行列式的方法很多，但具体到一个题，要针对其特征，选取适当的方法求解。当看到一个貌似非常复杂的n 阶行列式时，
仔细观察，
会发现其实它们的元素在行或列的排列方式上都有某些规律。掌握住这些规律，选择合适的计算方法，能使我们在极短的时间内达到事半功倍的效果！本文首先介绍n 阶行列式的定义、性质，再归纳总结行列式的各种计算方法、技巧及其在线性方程组中的初步应用。行列式是线性方程组理论的一个组成部分，是中学数学有关内容的提高和推广。它不仅是解线性方程组的重要工具，而且在其它一些学科分支中也有广泛的应用。
关键词：n 阶行列式 计算 方法 归纳
ABST RACT
线性方程组
Algebra is a courses of mathematics specialized compulsory of the basic mathematic. The determinant's calculation is the most difficulty in higher algebra, especially, the n order determinant's calculation , alway is student's difficulty in the learning process, so ,it is difficult to master for ours . There are a lot of calculations of
n order determinant in method , but when we say a problem of the calculation of n order determinant , according to its characteristics, selecting the appropriate method to solving is a very good idea. When you see a seemingly so complex n order determinant, we should observe them carefully,and we will find that their elements are arranged in rows or columns have some regularity. Grasping of these laws, finding a appropriate calculation method,it can help us to achieve a multiplier effect in a very short time! This paper mainly introduces the definition of n order determinant, nature, and calculation methods, the skills of calculation of n order determinant and application in linear equation group. Determinant is an important theory in linear equations and it is an indispensable part of linear equations , determinant is also the middle school mathematics' content raise and promotion. It is not only the solution of linear equations of the important tool, but also in some other branch has a wide range of applications.
Key words: n order determinant calculation method induce linear equations
目 录
引言„„„„„„„„„„„„„„„„„„„„„ 1 1 n 阶行列式的定义„„„„„„„„„„„„„„„ 3
2 n 阶行列式的性质„„„„„„„„„„„„„„„ 3 3 计算n 阶行列式的具体方法与技巧„„„„„„„„ 4
3.1 利用行列式定义直接计算„„„„„„„„„„ 4
3.2 利用行列式的性质计算„„„„„„„„„„„ 5
3.3 化为三角形行列式„„„„„„„„„„„„„ 6
3.4 降阶法„„„„„„„„„„„„„„„„„„ 7
3.5 逆推公式法„„„„„„„„„„„„„„„„ 8
3.6 利用范德蒙德行列式„„„„„„„„„„„„ 9
3.7 加边法（升阶法）„„„„„„„„„„„„„ 9
3.8 数学归纳法„„„„„„„„„„„„„„„„ 10
3.9 拆开法„„„„„„„„„„„„„„„„„„ 11 4 行列式在线性方程组中的初步应用„„„„„„„ 11
4.1 克拉默（Gramer ）法则„„„„„„„„„„„ 12
4.2 克拉默（Gramer ）法则的应用„„„„„„„„ 12
4.2.1 用克拉默（Gramer ）法则解线性方程组„„„ 13
4.2.2 克拉默法则及其推论在几何上的应用„„„„ 14 结论„„„„„„„„„„„„„„„„„„„„„ 16 参考文献„„„„„„„„„„„„„„„„„„„ 17 致谢„„„„„„„„„„„„„„„„„„„„„ 18
引 言
解方程是代数中一个基本问题，特别是在中学中所学的代数中，解方程占有重要的地位. 因此这个问题是读者所熟悉的. 比如说，如果我们知道了一段导线的电阻r ，它的两端的电位差v ，那么通过这段导线的电流强度i ，就可以有关系式
ir =v
求出来. 这就是所谓解一元一次方程的问题. 在中学所学代数中，我们解过一元、二元、三元以至四元一次方程组.
线性方程组的理论在数学中是基本的也是重要的内容.
对于二元线性方程组
⎧a x +a x =b ⎨1111221，
⎩a 21x 2+a 22x 2=b 2
当a 11a 22-a 12a 21≠0时，次方程组有惟一解，即 x 1=b 1a 22-a 12b 2a b -a b ， x 2=112121. a 11a 22-a 12a 21a 11a 22-a 12a 21
我们称a 11a 22-a 12a 21为二级行列式，用符号表示为
a 11a 22-a 12a 21=a 11a 12
a 21a 22
于是上述解可以用二级行列式叙述为：当二级行列式
时，该方程组有惟一解，即 a 11a 12a 21a 22≠0
b 1a 12
x 1=a 11b 1b 2a 22a b , x 2=212. a 11a 12a 11a 12
a 21a 22a 21a 22
对于三元线性方程组有相仿的结论. 设有三元线性方程组
⎧a 11x 1+a 12x 2+a 13x 3=b 1, ⎪ ⎨a 21x 1+a 22x 2+a 23x 3=b 2,
⎪a x +a x +a x =b . 3333⎩311322
称代数式a 11a 22a 33+a 12a 23a 31+a 13a 21a 32-a 11a 23a 32-a 12a 21a 33-a 13a 22a 31为三级行列式，用符号表示为：
a 11a 12a 13
a 11a 22a 33+a 12a 23a 31+a 13a 21a 32-a 11a 23a 32-a 12a 21a 33-a 13a 22a 31=a 21a 22a 23.
a 31a 32a 33
我们有：当三级行列式
a 11a 12a 13
d =a 21a 22a 23≠0
a 31a 32a 33
时，上述三元线性方程组有惟一解，解为 x 1=其中 d d 1d ，x 2=2，x 3=3 d d d
a 11b 1a 13
a 31b 3a 33a 11a 12b 1a 31a 32b 3b 1a 12a 13b 3a 32a 33 d 1=b 2a 22a 23 ，d 2=a 21b 2a 23，d 3=a 21a 22b 2
在本论文中我们将把这个结果推广到n 元线性方程组
⎧a 11x 1+a 12x 2+ +a 1n x n =b 1⎪a x +a x + +a x =b ⎪2112222n n 2 ⎨ ⎪⎪⎩a n 1x 1+a n 2x 2+ +a nn x n =b n
的情形. 为此，我们首先要给出n 阶行列式的定义并讨论它的性质，这就是本论文的主要内容.