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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:参考学习行列式的计算摘要：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(1704-1752)在其著作
2、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1n级行
3、列式a11a12a1n(1)a21a22a2naa*an1an2ann等于所有取自不同行不同列的n个元素的乘积a“a2j2anjn的代数和.其中j jn是1,2川l,n的一个排列，31j132j/' anjn的每一项都按下列规则带有符号:当jjjn是偶排列时，务朋22anjn带有正号，当jjjn是奇排列时,aije2j2anjn带有负号.1.2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21由s n个数排成的s行（横的）n列（纵的）的表an 1ai 1 a 1 2 I I）a2 1 a 2 2 HI*
4、f* frr.as1 as2 111称为一个s n矩阵.特别地，当s = n时，（1）称为（2）的行列式，如果把（2）记作A，贝U（1）表示为A .1定义3在行列式ana219an1312a22aan231 na2naann中划去元素aj所在的第i行和第j列后，剩下的（n -1）2个元素按照原来的排法构成一个n -1级行列式311aa1,j 1*Q'j 1a31na3i 4,13i 4, j 43i -1,j 13i,nai 1,1a3i 1,j 4ai 1,j 1aai 1 ,nan13n,j 43n,j 13nn(3)称为元素的余子式，记作Mj，而（-1）i jMj称为的代数余子式
5、，记作:(4)Aj =(T)，jMj定义4我们把(5)a 2 1川a 2 2川*+a2n川as as2-asn “称为矩阵（2）转置，记作A或A，显然，s n矩阵的转置是n s矩阵.的交点上的k2个元素按照原来的次序组成一个k级行列式M，称为行列式D的定义5 在一个n级行列式D中任意选定k行k列（k < n）位于这些行和列参考学习一个k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
6、不变，即a11a12aa1naa11a12aa1nai1ai2aainaai1 *cak1ai2 * Cak2aain * Ca knak1ak2aaknaak1ak2aaknan1an2a nnan1an2a nn(6)其中R.2）行列互换，行列式值不变，即ana12a1nana21an1a21I-a22aa2n9=a12I-a22an2aan1an2anna1na2nann(7)那么它就等于两个行列式的和，即,这两31 131 n31 2b1C1b2 C2bn cb1b2a n1a n2an na n1a n2r3）如果行列式的某一行是两组数的和，个行列式除这一行外其余与原来行列式对应相同
7、(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a11a a lia12-a1 naa11- 62aa1 nmkai1akai2 -kana=kai1-ai2aainman1ctali62annan1c 62ann性质4行列式反号对换行列式两行的位置，行列式反号(9)anB-a12aa1naania12 a1naai1ai2 ainak1ak2 aknfi-aa=iaak1B-ak2akn9ai1ai2aain9an1an2anna n1an2a nn(10)参考学习(11)(12)(13)(14)(15)H，" j；A,i=
8、jH，iH j3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A=(aj)为n级方阵，Aj为aj的代数余子式，则"A,i = jai1 Aj1ai2 Ajain AjnaiiAja2iA2janiAnj设A为n级方阵，则AA设A为n级方阵，则kA =kn A设A, B为n级方阵，则AB =|AB，但 A士B| M|A士 BAB = A B =囘 A = BA ,(但一般地 AB 式 BA)(16)(5)(拉普拉斯定理)设在n级行列式D中任意取定了 k(1 _ k _ n 一1)个行，由这k行元素所组成的一切k级子式与它们的代数余子式的乘积的和等于行列式参考学
9、习Am *Am 00Bn*BnB为n级方阵，则:=Am Bn ,0th A Bn(17)(7)范德蒙德行列式11IIIX1X2IIIDn =2片+2X2IH+n A.n A.X2IHAmBn01Xn2Xnn人 * III *九1ri*r1 *q*打打*川扎nXn(8) 一些特殊行列式的值(18)(19)设A为m级方阵,但是:对角行列式上三角行列式下三角行列式* III * 人:人2卡崩*frf- FK讯川*次对角行列式次上三角行列式(20)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计算行列式解：可以直接按照定义
10、把行列式写开，得D3 =2(x y)( -X2xy - y2) = -2(x3y3).322利用三角化法例2计算行列式1 -1 2D3 = 321014解：利用三角化法:1-121-121 -121-12D3 =321=05-5=(-5)0 -11=(-5)0-11=250140140 14005kJ3.3 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3.3.6数学归纳法3.3.7利用公式和定理参考学习参考文献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，1988. 03.2 张禾瑞
11、，郝炳新.高等代数M.北京高等教育出版社,1983. 04.3 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，2005. 09.4 耿锁华.行列式性质的应用M.南京审计学院出版社,2006. 01. 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，2007. 06. 刘崇华.一类行列式的计算公式M.南宁大学出版社，2006. 04.7 杨立英，李成群.n级行列式的计算方法与技巧M.广西师范学院出版社2006. 01.8 孙清华，孙昊，李金兰.高等代数内容、方法与技巧M.华中科技大学出版社,2006. 08.9 毛纲源.线性代数解题方法技巧归纳（第二版）M.华
12、中理工大学出版社2007. 06.The calculation of determinantAbstractDeterm inant is an importa nt tool to study in higher algebra. In this paper, from the determ inant calculatio n by examples, in troduces some methods of determ inant computati on, at the same time, the prelim inary calculati on method is give n
13、. Some special determ inant, draw some about the determ inant calculati on skills.KeywordsDetermi nant; tria ngulati on; factorizati on theorem; recursive method; mathematical in duct ionIntroductionSolv ing the determ inant in lin ear equati on s, it is the first expressi on is a shortha nd, now is
14、 a very useful tool in mathematics. The determ inant is inven ted by Leib niz and the Japa nese mathematicia n Seki takakazu. Con temporary Japa nese mathematicia n Seki Takakazu in his book "V" thematic method soluti on also proposed the con cept and algorithm of determ inant.In 1750, the
15、 Swiss mathematicia n Cramer (1704-1752) in his book "li near algebra an alysis guide", the defi niti on of the determ inant and expa nsion gives a relatively complete, clear, and gives now we call the solution of linear equations of the Cramer's rule. Later, the mathematicianBei Zu (1
16、7 30-1783) will determ ine the method of determ inant each symbol is a systematic con cept, using the coefficie nt determ inant points out how to judge a homoge neous lin ear equati ons with non-zero soluti on.The determ inant is one branch of mathematics as a tool, we learn in "Higher Algebra&
17、quot;, the book describes only the determ inant of some simple calculatio n methods, but in the face of the complicated or skills relatively strong determ inant, several methods are confined to the book, the problem a bit of trouble. Here I discuss some methods for calculati ng determ inant, the det
18、erm inant to choose accord ing to differe nt method to calculate the relative simple, to improve the efficie ncy of problem solv ing.1 A brief introduction to the Basic Concepts1.1 n determinantDefines 1 levels of determi nanta11a12a1na21a22a2n(1)an1an2annIs equal to the algebraic sum of all taken f
19、rom different lines of different column n elements of the productaa2j2a% Where 訂2jn is the 1, 2,n an order,玄门卫？)？a% each one of them according to the following rules with symbols: when h j2 j n is even permutation, with positive a1 j1a2j1 anjn ,訂2" jn when is odd permutation, a1j1a2jL anjn with
20、 a minus sig n.1.2 matrixMay be used as matrix and its related con cept in the process of the determ inant of the determ inant formula and con clusi ons and back calculati on, so here is simple to in troduce the con cept of matrix and its parts.s n into s lines (horizontal) n column (vertical) inDef
21、inition 2 by the nu mber ofa1 1Tablea2 1(2)as2川Known as a s n matrix.In particular, whens 二 n, (1) (2) is called the determinant, if (2) denoted as A, then (1) expressed as a ADefinition 3In the determ inant ofa11a21a12a22a1na2nan1an2annIn return for element ajin the i2and j columns, the rest of the
22、 (n1) elementsanaa1,j Jaa1,j 1-aa1najai 1,19ai J, j 1ai 1, j 1aid,j-1-ai 1,j 1ai,nai 1,nan1an,j Aan,j 1-ann(3) Known as the cofactorelement ajtype, denoted as M j, while the (-1) jMj iscalled the algebraic ajtype, deno ted as:accord ing to the orig inal method con sisti ng of a n-1 determ inant ofAj
23、 -(-1)i jMjDefinition 4 We call3 1a1 2asas(5)a2nIIIasn对Known as the matrix transpose (2), denoted as A or AT, apparently, transpose of s n matrix is n s .Definition 5 In n determ inant of D in any of the selected k row k colum n (k _ n) is2located in the intersection of these rows and columns of the
24、 k elements according to the origi nal order in which a k determ inant of M, called a k step determ inant of D type.2 Properties of the determinantAccord ing to the value of determ inant can be divided into the follow ing categories: Properties of determinant value is 01) If there are two lines of t
25、he same determi nant, the determ inant value of 0;2) If the determ inant is two in proporti on, the determ inant value of 0;3) The determ inant of a behavior of 0, the determ inant of the value of 0(2) Properties of determinant.(10)1) The line ratio to ano ther line, the determ inant of inv aria nt,
26、 i.e.ai1 ai2ai1 ai2ak1 ak2a n1 a n2a1naa11a12aa1nainaai1 +cak1ai2 + Cak2 aain * Caknaknaak1I-ak2aknaa nna n1an2ann参考学习2) Tran spose, determ inant value un cha nged, i.e.a11ai2a1 na11a21an1a21-a22-a2n=a12a22aan2aan1an2anna1na2nann(7)is equal to the two3) If a row determ inant is two sets of nu mbers
27、and, the n it determ inant and the two determ inan t, i n additi on to the line outside the rest with theorig inal determ inant corresp onding to the same, i.e.ana12a1 na11 a 12a 1na11 a12a1nb1 亠 C1b2 亠 C2bn +Cn = b1 b2bn + C1C2(8)an1a n2anna n1 a n2a nna n1 an2a nn(3) Change properties of determina
28、ntThe com mon factor line can be put forward to, or use a multiplied by the determ inant of a is equal to the nu mber is multiplied by this determ inanta11aa12aa1 naa11费a129a1nakai1aka2akaina=kai1费ai29aina(9)an1an2annan1an2ann(4) Properties of determinant inverse numberOn line two, the nu mber of de
29、term inanta11aa12-aa1naai1aai2 -aainaak1aak2'aaknaan1an2 "annai1ai2 ainak1 ak2aknail ai2 aina n1 a n2a nn3Calculation of determinant3.1 Some important formulas and conclusions(1) The determ inant line (or colu mn) expa nsionLet A =佝)be n matrix, the A cofactor aj type, then A,i = jai1 Aj! +
30、 恥 Aj2 + ain Ajn =川(11).0,1 和A i = jaAj +a2iA2j + 七帘人可(12)卫心j(2) Let A be a n matrix,At| = A(13)(3) Let A be a n matrix,kA = knA(14) Let A, B is n matrix, AB =|A B ,但 A± B 鼻 A ±|B (15)AB| |A B| |B A 二 BA , (, but generallyAB = BA )(16)(5) (Laplasse theorem) In arbitrary n determ inant of D
31、 in thek(1 乞 k 乞 n 1)line,product of algebraic all k type con sisted of the k eleme nts and their type and is equal to the determ inant of D.(6) Let A be a m matrix, B matrix, n,BnAm0BnAn BnBut, ABn =(Tmn|Arn|Bn|( 17)Am 0(18).(7) Van Redm ond determ inant* III *'-2f=ft*為：*曲=*社2ip*t* III*打為*川*人=*
32、+i=卡i扎2*F%*rK *III *(8) Some special determ inant value(19)(20).11IH1xX2IIIXnDn =2X；IN2Xnn(xj - xi)+dFb1 兰+IHFx1x2xn参考学习Notes: (19) (20) of the determ inant of the eleme nts * are not all zero.3.2Calculation of primary determinant3.2.1 Use of the definition of the determinant, properties1 casesof co
33、mputi ng determ inant ofx y x + yD3 = y x + y xx + y x ySoluti on: can be directly accord ing to the defi niti on of the determ inant is writte n, D3 =2(x y)(-x2 xy - y2) = -2(x3 y3).3.2.2 Uses triangulation method2 casesof computi ng determ inant of1-12D3= 321014Solution the use of triangulation me
34、thod1-121-121-121-12D3 =321=05_ 5=(-5)0-11=(-5)0-11=250140140140053.3Calculation of level n determinant3.3.1Using the definition3.3.2 Row (column) subtract (add) method3.3.3 Factor theorem method3.3.4The recursive degradation method3.3.5 Method3.3.6 Mathematical induction3.3.7Using the formula and t
35、heorem参考学习Reference1 Wang Efan g, Ihi Kim algebraic geometry and higher algebra M. Departme nt of Pek ing Un iversity Departme nt of mathematics of algebra group cod ing,1988.03.2 Zhang Herui, Hao Bingxin. Advaneed algebra M. Beijing higher education press, 1983.04.3 Li Zhihui, Li Yongming and M. Of
36、 Shaanxi Normal University College of mathematics and in formatio n scie nee of higher algebra,2005.09.4 Geng Suohua. The determ inant of the n ature of the applicati on M. Nanjing Audit University press, 2006.01.5 Korea, Guo Haiqi ng. Calculatio n of M. Southwest Un iversity for Nati on alities pre
37、ss two kinds of special li ne history, 2007.06.6 Liu Chonghua. A class of determinantai formula for M. Nanning University Press, 2006.04.7 Yang Liying, Li Chengqun. Calculation methods and skills of primary determinant M. Guangxi Teachers Education University press, 2006.01.8 Sun Qin ghua, Sun Hao, Li Jinlan and skill of Higher Algebra content, method of M. Huazh ong Uni versity of Scie nee and Tech no logy press, 2006.08.9 Hair Gan gyua n lin ear algebra problem solvi ng methods. Tech niq ues (Sec ond Editi on) M. Huazho ng Uni versity of scie nee and Tech no logy Press, 2007.06.