1、 X X 大 学行列式的计算学生姓名： 学 号： 班 级： 专 业：系 别: 指导教师： 行列式的计算摘 要：行列式是高等代数研究中的一个重要工具.本文从行列式的计算出发，通过例题，介绍行列式计算中的一些方法，同时初步给出了一些特殊行列式的计算方法，得出了一些关于行列式计算的技巧.关键词：行列式；三角化法；因式定理法；递推法；数学归纳法 引 言行列式出现于线性方程组的求解，它最早是一种速记的表达式，现在已经是数学中一种非常有用的工具.行列式是由莱布尼茨和日本数学家关孝和发明的.同时代的日本数学家关孝和在其著作解伏题元法中也提出了行列式的概念与算法.1750 年，瑞士数学家克拉默(1704-17
2、52)在其著作线性代数分析导引中，对行列式的定义和展开法则给出了比较完整、明确的阐述，并给出了现在我们所称的解线性方程组的克拉默法则.稍后，数学家贝祖 (1730-1783)将确定行列式每一项符号的方法进行了系统化，利用系数行列式概念指出了如何判断一个齐次线性方程组有非零解.行列式是多门数学分支学科一个工具，在我们学习高等代数时，书中只介绍了几种较简单的行列式计算方法，但是在遇到比较复杂或技巧性比较强的行列式时，只局限于书上的几种方法，那解题就有点麻烦.这里我讨论了行列式计算的若干方法，针对不同的行列式来选择相对简单的计算方法，来提高解题的效率.1 基本概念的简单介绍1.1 n 级行列式定义
3、1 级行列式（1）nnnaa 212112等于所有取自不同行不同列的 个元素的乘积 的代数和.其中njja21是 的一个排列， 的每一项都按下列规则带有符号：nj21, njja21当 是偶排列时， 带有正号，当 是奇排列时， njj21 nj21带有负号.njja211.2 矩阵在叙述行列式的重要公式和结论以及后面计算行列式过程中可能要用到矩阵及其有关概念，所以在这里简单介绍一下矩阵及其部分概念.定义 2 由 个数排成的 行（横的） 列（纵的）的表1snsn121212nssnaa（2）称为一个 矩阵.sn特别地，当 时， （1）称为（2）的行列式，如果把（2）记作 ，则 A（1）表示为 .
4、A定义 3 在行列式1 nnnaa 212112中划去元素 所在的第 行和第 列后，剩下的 个元素按照原来的排法ijaij 2)1(n构成一个 级行列式1n（3）njnjnn ijijii njj aaaa 1,1,1 ,1, 111 1, 称为元素 的余子式，记作 ，而 称为 的代数余子式，记作：ijaijMiji)(ij（4）ijiijA1定义 4 我们把1（5121212snsnaa）称为矩阵（2）转置，记作 或 ，显然， 矩阵的转置是 矩阵.AT ns定义 5 在一个 级行列式 中任意选定 行 列 位于这些行和列1nDk)(的交点上的 个元素按照原来的次序组成一个 级行列式 ,称为行列
5、式 的2k MD一个 级子式.2 行列式的性质按照行列式的值可分为以下几类：性质 1 行列式值为 01) 如果行列式有两行相同，则行列式值为 0;2) 如果行列式有两行成比例，则行列式值为 0;3) 行列式中有一行为 0，则行列式的值为 0.性质 2 行列式值不变1) 把一行的倍数加到另一行，行列式值不变, 即（6）nnn kkkknikikinnkkinii aacacaaaaa 2121 1121211 其中 .Rc2) 行列互换，行列式值不变, 即= （7）nnnaa 212112 nnna 2121213) 如果行列式的某一行是两组数的和，那么它就等于两个行列式的和, 这两个行列式除这
6、一行外其余与原来行列式对应相同,即 nnnnnnnnn n aaccabbaaccb 2111221112211121 （8）性质 3 行列式的值改变一行的公因子可以提出去，或者说用一数乘以行列式的一行就等于用该数乘以此行列式（9）nniiinnniii naaaakaka 2111221112性质 4 行列式反号对换行列式两行的位置，行列式反号（10）nniiiknknnkkinii aaaaaa 21211212113 行列式的计算3.1 一些重要的公式和结论(1) 行列式按行（或列）展开设 为 级方阵， 为 的代数余子式，则)(ijaAnijAija（11）jiAjnijiji ,021
7、（12）jiaAanjijiji ,21(2) 设 为 级方阵，则An（13）T(3) 设 为 级方阵，则n（14）Akn(4) 设 为 级方阵，则BA,n,但 （15）BABA, （但一般地 ） BA（16）(5) (拉普拉斯定理)设在 级行列式 中任意取定了 个行，由nD)1(nk这 行元素所组成的一切 级子式与它们的代数余子式的乘积的和等于行列式kk.D(6) 设 为 级方阵， 为 级方阵，则：AmBn,00mmnnnAAB但是：（17）0(1)nmnmBABA(7) 范德蒙德行列式（18）122112()nn jiijnnnxxDx(8) 一些特殊行列式的值（19）111222nnn
8、对角行列式 上三角行列式 下三角行列式 （20）111222nnn 次对角行列式 次上三角行列式 次下三角行列式 说明：（19） （20）中的行列式中*号处的元素不全为零. 3.2 低级行列式的计算 3.2.1 利用行列式定义，性质例 1 计算行列式 yxyD3解：可以直接按照定义把行列式写开，得.)(2)(2323 yxyxyx3.2.2 利用三角化法例 2 计算行列式 31204D解：利用三角化法：.410524231D12()0412(5)053.3 n 级行列式的计算3.3.1 利用定义3.3.2 逐行（列）相减（加）法3.3.3 利用因式定理法3.3.4 递推降级法3.3.5 拆分法
9、3.3.6 数学归纳法3.3.7 利用公式和定理参考文献1 王萼芳，石生明高等代数M北京大学数学系几何与代数教研室前代数小组编, 1988032 张禾瑞，郝炳新高等代数M北京高等教育出版社, 1983043 李志慧，李永明高等代数分析与选讲M陕西师范大学数学与信息科学学院, 2005094 耿锁华行列式性质的应用M南京审计学院出版社, 2006015 高丽，郭海清两类特殊行历史的计算M西南民族大学出版社, 2007066 刘崇华 一类行列式的计算公式M南宁大学出版社, 200604.7 杨立英，李成群 级行列式的计算方法与技巧M 广西师范学院出版社, n200601.8 孙清华，孙昊，李金兰高
10、等代数内容、方法与技巧M华中科技大学出版社, 200608.9 毛纲源线性代数解题方法技巧归纳(第二版)M华中理工大学出版社, 200706.The calculation of determinantAbstract Determinant is an important tool to study in higher algebra. In this paper, from the determinant calculation by examples, introduces some methods of determinant computation, at the same time
11、, the preliminary calculation method is given. Some special determinant, draw some about the determinant calculation skills.KeywordsDeterminant; triangulation; factorization theorem; recursive method; mathematical inductionIntroductionSolving the determinant in linear equations, it is the first expr
12、ession is a shorthand, now is a very useful tool in mathematics. The determinant is invented by Leibniz and the Japanese mathematician Seki takakazu. Contemporary Japanese mathematician Seki Takakazu in his book “V“ thematic method solution also proposed the concept and algorithm of determinant.In 1
13、750, the Swiss mathematician Cramer (1704-1752) in his book “linear algebra analysis guide“, the definition of the determinant and expansion gives a relatively complete, clear, and gives now we call the solution of linear equations of the Cramers rule. Later, the mathematician Bei Zu (17 30-1783) wi
14、ll determine the method of determinant each symbol is a systematic concept, using the coefficient determinant points out how to judge a homogeneous linear equations with non-zero solution.The determinant is one branch of mathematics as a tool, we learn in “Higher Algebra“, the book describes only th
15、e determinant of some simple calculation methods, but in the face of the complicated or skills relatively strong determinant, several methods are confined to the book, the problem a bit of trouble. Here I discuss some methods for calculating determinant, the determinant to choose according to differ
16、ent method to calculate the relative simple, to improve the efficiency of problem solving.1 A brief introduction to the Basic Concepts1.1 n determinantDefines 1 levels of determinant（1）nnnaa 212112Is equal to the algebraic sum of all taken from different lines of different column n elements of the p
17、roduct Where is the 1, 2, n an order, each one of njja21 nj21 njja21them according to the following rules with symbols: when is even permutation, with nj21positive , when is odd permutation, with a minus sign.njj21 nj21 njja211.2 matrixMay be used as matrix and its related concept in the process of
18、the determinant of the determinant formula and conclusions and back calculation, so here is simple to introduce the concept of matrix and its parts.Definition 2 by the number of into s lines (horizontal) n column (vertical) in snTable （2）12112nssaaKnown as a matrix.nIn particular, when , (1) (2) is
19、called the determinant, if (2) denoted as A, then (1) sexpressed as a ADefinition 3In the determinant of nnnaa 212112In return for element in the i and j columns, the rest of the elements ij 2)1(naccording to the original method consisting of a n-1 determinant of （3）.njnjnn ijijii njj aaaa 1,1,1 ,1,
20、 111 1, Known as the cofactor element type, denoted as , while the is ij ijMijiM)1(called the algebraic type, denoted as: ija（4）.ijiijA)1(Definition 4 We call （5）121212snsnaaKnown as the matrix transpose (2), denoted as or , apparently, transpose of ATmatrix is .snsDefinition 5 In n determinant of D
21、 in any of the selected k row k column is )(nlocated in the intersection of these rows and columns of the elements according to 2the original order in which a k determinant of M, called a k step determinant of D type.2 Properties of the determinantAccording to the value of determinant can be divided
22、 into the following categories:(1) Properties of determinant value is 01) If there are two lines of the same determinant, the determinant value of 0;2) If the determinant is two in proportion, the determinant value of 0;3) The determinant of a behavior of 0, the determinant of the value of 0(2) Prop
23、erties of determinant.1) The line ratio to another line, the determinant of invariant, i.e.（6）nnn kkkknikikinnkkinii aacacaaaaa 2121 1121211 Rc2) Transpose, determinant value unchanged, i.e.= （7）nnnaa 212112 nnna 2121213) If a row determinant is two sets of numbers and, then it is equal to the two d
24、eterminant and the two determinant, in addition to the line outside the rest with the original determinant corresponding to the same, i.e.（8）nnnnnnnnn n aaccabbaaccb 2111221112211121 (3)Change properties of determinantThe common factor line can be put forward to, or use a multiplied by the determina
25、nt of a is equal to the number is multiplied by this determinant（9）nniiinnniii naaakaka 2111221112(4)Properties of determinant inverse numberOn line two, the number of determinant（10）nniiiknknnkkinii aaaaaa 21211212113Calculation of determinant3.1 Some important formulas and conclusions(1) The deter
26、minant line (or column) expansionLet be n matrix, the cofactor type, then)(ijaAijAija（11）jiajnijiji ,021（12）jiAAanjijiji ,21(2) Let A be a n matrix, （13）T(3) Let A be a n matrix, （14）Akn(4) Let A, B is n matrix, ,但 （15）BB, (, but generally ) (16)AA A(5) (Laplasse theorem) In arbitrary n determinant
27、of D in the line, )1(nkproduct of algebraic all k type consisted of the k elements and their type and is equal to the determinant of D.(6) Let A be a m matrix, B matrix, n,00mnnnABBut,（17）(1)nmnmB(7) Van Redmond determinant(18).122112()nn jiijnnnxxDx(8) Some special determinant value(19)111222nnn (2
28、0).111222nnn Notes: (19) (20) of the determinant of the elements * are not all zero.3.2Calculation of primary determinant3.2.1 Use of the definition of the determinant, properties1 cases of computing determinant of yxyD3Solution: can be directly according to the definition of the determinant is writ
29、ten, .)(2)(2323 yxyxyx3.2.2 Uses triangulation method2 cases of computing determinant of 31204DSolution: the use of triangulation method 410524231D12()0412(5)053.3Calculation of level n determinant3.3.1Using the definition 3.3.2 Row (column) subtract (add) method3.3.3 Factor theorem method3.3.4The r
30、ecursive degradation method3.3.5 Method3.3.6 Mathematical induction3.3.7Using the formula and theoremReference1 Wang Efang, Ihi Kim algebraic geometry and higher algebra M. Department of Peking University Department of mathematics of algebra group coding,1988.03.2 Zhang Herui, Hao Bingxin. Advanced
31、algebra M. Beijing higher education press, 1983.04.3 Li Zhihui, Li Yongming and M. Of Shaanxi Normal University College of mathematics and information science of higher algebra,2005.09.4 Geng Suohua. The determinant of the nature of the application M. Nanjing Audit University press, 2006.01.5 Korea,
32、 Guo Haiqing. Calculation of M. Southwest University for Nationalities press two kinds of special line history, 2007.06.6 Liu Chonghua. A class of determinantal formula for M. Nanning University Press, 2006.04.7 Yang Liying, Li Chengqun. Calculation methods and skills of primary determinant M. Guang
33、xi Teachers Education University press, 2006.01.8 Sun Qinghua, Sun Hao, Li Jinlan and skill of Higher Algebra content, method of M. Huazhong University of Science and Technology press, 2006.08.9 Hair Gangyuan linear algebra problem solving methods. Techniques (Second Edition) M. Huazhong University of science and Technology Press, 2007.06.
