线性代数1.6方阵的行列式.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,第一章 矩阵,1.6,方阵的行列式,回忆,:,1.5,一开始提出的问题,.,习题,1(B),第,17,题,:,a,11,a,12,a,21,a,22,A,=,可逆,一阶方阵,a,可逆,a,0.,a,11,a,22,a,12,a,21,0,a,11,a,12,a,21,a,22,D,=,0.,第一章 矩阵,1.6,方阵的行列式,1.6,方阵的行列式,历史上,行列式因线性方程组的求解而被发明,G.W.,Leibniz,德,(,1646.7.1,1716.11.14,),S.,Takakazu,日,(,1642?,1708.10.24,),第一章 矩阵,1.6,方阵的行列式,(,a,11,a,22,a,12,a,21,),x,1,=,b,1,a,22,a,12,b,2,(,a,11,a,22,a,12,a,21,),x,2,=,a,11,b,2,b,1,a,21,当,a,11,a,22,a,12,a,21,0,时,a,11,x,1,+,a,12,x,2,=,b,1,a,21,x,1,+,a,22,x,2,=,b,2,x,1,=,b,1,a,22,a,12,b,2,a,11,a,22,a,12,a,21,x,2,=,a,11,a,22,a,12,a,21,a,11,b,2,b,1,a,21,.,消元法,由方程组的四个系数确定,.,由四个数排成二行二列（横排称行、竖排,称列）的数表,定义,即,一,.,行列式,(,determinant,),的定义,主,对角线,副对角线,对角线法则,二阶,行列式的计算,若记,对于二元线性方程组,系数行列式,第一章 矩阵,1.6,方阵的行列式,a,11,a,12,a,21,a,22,记,D,=,b,1,a,12,b,2,a,22,D,1,=,a,11,b,1,a,21,b,2,D,2,=,则当,D,=,a,11,a,22,a,12,a,21,0,时,=,D,1,D,=,D,2,D,.,a,11,x,1,+,a,12,x,2,=,b,1,a,21,x,1,+,a,22,x,2,=,b,2,x,1,=,b,1,a,22,a,12,b,2,a,11,a,22,a,12,a,21,有唯一确定的解,x,2,=,a,11,a,22,a,12,a,21,a,11,b,2,b,1,a,21,例1,解,二、三阶行列式,定义,记,（,6,）式称为数表（,5,）所确定的,三阶行列式,.,(1),沙,路法,三阶行列式的计算,.,列标,行标,(2),对角线法则,注意,红线上三元素的乘积冠以正号，蓝线上三,元素的乘积冠以负号,说明,1,对角线法则只适用于二阶与三阶行列式,第一章 矩阵,1.6,方阵的行列式,例,2,.,1 2 4,2 2 1,3 4 2,=14.,第一章 矩阵,1.6,方阵的行列式,一般地,在,n,阶行列式中,把元素,a,i,j,所在的第,i,行,和第,j,列划去,留下来的,n,1,阶行列式叫做元素,a,i,j,的,余子式,(,minor,),记作,M,i,j,令,A,i,j,=(,1,),i+,j,M,i,j,并称之为,a,i,j,的,代数余子式,(cofactor).,例如,四阶阶行列式,中,a,32,的余子式为,a,11,a,12,a,13,a,14,a,21,a,22,a,23,a,24,a,31,a,32,a,33,a,34,a,41,a,42,a,43,a,44,a,11,a,13,a,14,a,21,a,23,a,24,a,41,a,43,a,44,M,32,=,代数余子式,A,32,=(,1,),3+2,M,32,=,M,32,.,a,11,a,12,a,13,a,21,a,22,a,23,a,31,a,32,a,33,第一章 矩阵,1.6,方阵的行列式,a,11,a,12,a,13,a,21,a,22,a,23,a,31,a,32,a,33,a,11,的,余子式,:,a,22,a,23,a,32,a,33,M,11,=,代数余子式,:,A,11,=(1),1+1,M,11,a,12,的,余子式,:,a,21,a,23,a,31,a,33,M,12,=,代数余子式,:,A,12,=(1),1+2,M,12,a,13,的,余子式,:,M,13,=,代数余子式,:,A,13,=(1),1+3,M,13,a,21,a,22,a,31,a,32,a,11,a,12,a,13,a,21,a,22,a,23,a,31,a,32,a,33,第一章 矩阵,1.6,方阵的行列式,3,阶方阵,A,=,的行列式,|,A,|,定义为,a,11,a,12,a,13,a,21,a,22,a,23,a,31,a,32,a,33,|,A,|=,a,11,a,12,a,13,a,21,a,22,a,23,a,31,a,32,a,33,=,a,11,A,11,+,a,12,A,12,+,a,13,A,13,=,a,11,a,22,a,33,+,a,12,a,23,a,31,+,a,13,a,21,a,32,a,11,a,23,a,32,a,12,a,21,a,33,a,13,a,22,a,31,.,第一章 矩阵,1.6,方阵的行列式,补充,.,数学归纳法,(Principle of mathematical induction),1.,第一数学归纳法原理,:,则,P,对于任意的自然数,n,n,0,成立,.,设,P,是一个关于自然数,n,的命题,若,P,对于,n,=,n,0,成立,.,当,n,n,0,时,由“,n,=,k,时,P,成立”可推出,“,n,=,k,+1,时,P,成立”,第一章 矩阵,1.6,方阵的行列式,2.,第二数学归纳法原理,:,设,P,为一个关于自然数,n,的命题,若,P,对于,n,=,n,0,成立,由“,n,0,n,k,时,P,成立”可推出,“,n,=,k,+1,时,P,成立”,则,P,对于任意的自然数,n,n,0,成立,.,第一章 矩阵,1.6,方阵的行列式,a,11,a,12,a,1,n,a,21,a,22,a,2,n,a,n,1,a,n,2,a,nn,=,a,11,A,11,+,a,12,A,12,+,a,1,n,A,1,n,假设,n,1,阶行列式已经定义,=,a,11,(1),1+1,M,11,+,a,12,(1),1+2,M,12,+,a,1,n,(1),1+,n,M,1,n,n,1,阶行列式,(,Laplace,Expansion of Determinants),P.-S.,Laplace,法,(,1749.3.23,1827.3.5,),则定义,n,阶行列式,说明,1,、行列式是一种特定的算式，它是根据求解方程个数和未知量个数相同的一次方程组的需要而定义的,;,2,、阶行列式是 项的代数和,;,3,、阶行列式的每项都是位于不同行、不同列 个元素的乘积,;,4,、一阶行列式 不要与绝对值记号相混淆,;,5,、的符号为,第一章 矩阵,1.6,方阵的行列式,例,2,.,1 2 4,2 2 1,3 4 2,=14.,第一章 矩阵,1.6,方阵的行列式,例,3,.,下三角形,(,lower,triangular),行列式,a,11,0,0,a,21,a,22,0,a,n,1,a,n,2,a,nn,=,a,11,a,22,a,nn,.,例,4,.,上三角形,(upper triangular),行列式,a,11,a,12,a,1,n,0,a,22,a,2,n,0,0,a,nn,=,a,11,a,22,a,nn,.,第一章,(determinant),教学目的和要求：,1,、理解行列式的性质。,2,、掌握行列式的计算方法。,3,、理解伴随矩阵的定义及性质。,4,、了解行列式的应用。,本节重难点：,重点,是掌握行列式的计算方法,；伴随矩阵的定义及性质；,难点,是伴随矩阵的性质；,第六节 行列式（,2,）,第一章 矩阵,1.6,方阵的行列式,二,.,行列式的性质,性质,1,.,互换行列式中的两列,行列式变号,.,推论,.,若行列式,D,中有两列完全相同,则,D,=0.,a,11,a,12,a,21,a,22,例如,=,a,11,a,22,a,12,a,21,a,12,a,11,a,22,a,21,=,a,12,a,21,a,11,a,22,.,1,1,2,2,D,=,=,1,1,2,2,=,D,D,=0.,第一章 矩阵,1.6,方阵的行列式,性质,2,.(,线性性质,),(1)det(,1,k,j,n,),=,k,det(,1,j,n,);,(2)det(,1,j,+,j,n,),=det(,1,j,n,),+det(,1,j,n,).,现学现用,(1),设,A,为,n,阶方阵,则,det(,A,)=_,det(,A,).,(1),n,(2),a,+,b,c,+,d,u,+,v,x,+,y,=.,a,c,u,x,+,b,d,v,y,a,c,u,x,+,a,d,u,y,+,b,c,v,x,+,b,d,v,y,.,第一章 矩阵,1.6,方阵的行列式,推论,.,若行列式,D,中有两列元素成比例,则,D,=0.,a,11,a,1,i,k,a,1,i,a,1,n,a,21,a,2,i,k,a,2,i,a,2,n,a,n,1,a,n,i,k,a,n,i,a,nn,=,k,0,=0.,=,k,a,11,a,1,i,a,1,i,a,1,n,a,21,a,2,i,a,2,i,a,2,n,a,n,1,a,n,i,a,n,i,a,nn,例,第一章 矩阵,1.6,方阵的行列式,性质,3,.,把行列式的某一列的,k,倍加到另一列,上去,行列式的值不变,.,a,11,(,a,1,i,+,k,a,1,j,),a,1,j,a,1,n,a,21,(,a,2,i,+,k,a,2,j,),a,2,j,a,2,n,a,n,1,(,a,n,i,+,k,a,n,j,),a,n,j,a,nn,=,a,11,a,1,i,a,1,j,a,1,n,a,21,a,2,i,a,2,j,a,2,n,a,n,1,a,n,i,a,n,j,a,nn,+,a,11,k,a,1,j,a,1,j,a,1,n,a,21,k,a,2,j,a,2,j,a,2,n,a,n,1,k,a,n,j,a,n,j,a,nn,第一章 矩阵,1.6,方阵的行列式,例,1,.,1 2 4,2 2 1,3 4 2,(2),1 0 4,=-,2 6 1,3 10 2,1,0,0,=-,2,(,7,),2,3,1,3,5,2,1,0,0,=14 2,0,1,3,1,2,1,0,0,=-,14,2,1,0,3,2,1,=14.,4,1 0 0,=-,2 6 7,3 10 14,(3),注,:,本题也可以用定义或对角线法则计算,.,2 1 4,2 -2 1,4 -3 2,=-,第一章 矩阵,1.6,方阵的行列式,性质,4,.,设,A,B,为同阶方阵,则,|,AB,|=|,A,|,B,|.,性质,5,.|,A,T,|=|,A,|.,注,:,根据方阵的性质,5,前面几条关于,列,的,性质可以翻译到,行,的情形,.,例如,:,性质,1,.,互换行列式中的两,行,行列式变号,.,A.L.,Cauchy,法,(,1789.8.21,1857.5.23,),第一章 矩阵,1.6,方阵的行列式,定理,1.7,.,n,阶行列式,D,等于它的任意一行,(,列,),的各元素与其对应的代数余子式乘积,之和,.,即,D,=,a,1,1,A,1,1,+,a,1,2,A,1,2,+,a,1,n,A,1,n,=,a,2,1,A,2,1,+,a,2,2,A,2,2,+,a,2,n,A,2,n,=,=,a,n,1,A,n,1,+,a,n,2,A,n,2,+,a,n,n,A,n,n,=,a,1,1,A,1,1,+,a,2,1,A,2,1,+,a,n,1,A,n,1,=,a,1,2,A,1,2,+,a,2,2,A,2,2,+,a,n,2,A,n,2,=,=,a,1,n,A,1,n,+,a,2,n,A,2,n,+,a,n,n,A,n,n,.,第一章 矩阵,1.6,方阵的行列式,性质,6,.,a,i,1,A,j,1,+,a,i,2,A,j,2,+,a,in,A,jn,=0(,i,j,),a,1,i,A,1,j,+,a,2,i,A,2,j,+,a,ni,A,nj,=0(,i,j,).,定理,1.8,.,设,D,=|,a,ij,|,则,a,i,k,A,j,k,=,D,i,j,k,=1,n,a,k,i,A,k,j,=,D,i,j,.,k,=1,n,注,:,克罗内克记号,i,j,=,1,i,=,j,0,i,j,.,L.,Kronecker,德,(,1823.12.7,1891.12.29,),第一章 矩阵,1.6,方阵的行列式,三,.,行列式的计算,1.,二,三阶行列式,对角线法则,.,例,2,解,按,对角线法则，有,第一章 矩阵,1.6,方阵的行列式,2.,按某一行,(,列,),展开,降阶,.,例,2,例,3,计算,解,评注,本题是利用行列式的性质将所给行列,式的某行（列）化成只含有一个非零元素，然后,按此行（列）展开，每展开一次，行列式的阶数,可降低,1,阶，如此继续进行，直到行列式能直接,计算出来为止（一般展开成二阶行列式）这种,方法对阶数不高的数字行列式比较适用,练习,计算,（见,P52 25,（,3,）,第一章 矩阵,1.6,方阵的行列式,(,其中,n,2,x,a,).,D,n,=,x,a,a,a,x,a,a,a,x,例,4.,计算,n,阶行列式,3.,利用初等变换化为三角形,.,第一章 矩阵,1.6,方阵的行列式,D,n,=,x,a,a,a,x,a,a,a,x,x,+(,n,1),a,a,a,x,+(,n,1),a,x,a,x,+(,n,1),a,a,x,=,解,:,(,1,),x,+(,n,1),a,a,a,a,a,0,x,a,0 0 0,0 0,x,a,0 0,0 0 0 ,x,a,0,0 0 0 0,x,a,=,=,x,+(,n,1),a,(,x,a,),n,1,.,练习,计算,解法一,解法二,第一章 矩阵,1.6,方阵的行列式,4.,递推,/,归纳,.,(,未写出的元素都是,0).,例,5.,计算,2,n,阶行列式,D,2,n,=,a,b,a,b,c,d,c,d,第一章 矩阵,1.6,方阵的行列式,解,:,D,2,n,=,=a,.,.,.,.,.,.,.,.,.,.,.,.,a,a,b,b,0,c,c,0,d,d,0,0,d,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,0,a,a,b,b,c,0,c,c,0,d,d,0,.,.,.,+(,1,),2,n,+1,b,.,.,.,.,.,.,.,.,.,.,.,.,a,0,0,a,a,b,c,d,d,0,0,d,.,.,.,0,b,b,0,0,c,c,0,.,.,.,.,.,.,.,.,.,第一章 矩阵,1.6,方阵的行列式,=a,.,.,.,.,.,.,.,.,.,.,.,.,a,a,b,b,0,c,c,0,d,d,0,0,d,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,0,a,a,b,b,c,0,c,c,0,d,d,0,.,.,.,+(,1,),2,n,+1,b,=,ad D,2(,n,1,),bc,D,2(,n,1,),=(,ad,bc,),D,2(,n,1,),=(,ad,bc,),2,D,2(,n,2,),=(,ad,bc,),3,D,2(,n,3,),=(,ad,bc,),n,1,D,2,=(,ad,bc,),n,.,第一章 矩阵,1.6,方阵的行列式,例,6,.,证明,n,阶级,(,n,2),范德蒙行列式,D,n,=,1 1 1,a,1,a,2,a,n,a,1,2,a,2,2,a,n,2,a,1,n,-1,a,2,n,-1,a,n,n,-1,=(,a,i,a,j,).,n,i,j,1,Alexandre-Thophile,Vandermonde,Born:,28 Feb 1735 in Paris,France,Died:,1 Jan 1796 in Paris,France,第一章 矩阵,1.6,方阵的行列式,四,.,行列式的应用,设,A,=,a,ij,n,n,为方阵,元素,a,ij,的代数余子,式为,A,ij,则称如下矩阵,A,*,=,A,11,A,21,A,n,1,A,12,A,22,A,n,2,A,1,n,A,2,n,A,nn,为方阵,A,的,伴随矩阵,(,adjoint,).,1.,伴随矩阵与逆矩阵,第一章 矩阵,1.6,方阵的行列式,例,7,.,求,A,=,a,b,c,d,的伴随矩阵,.,解,:,A,11,=,d,A,21,=,b,A,12,=,c,A,22,=,a,.,A,*,=,A,11,A,21,A,12,A,22,=,d,b,c,a,.,第一章 矩阵,1.6,方阵的行列式,例,8,.,设,A,为方阵,A,*,为其伴随矩阵,.,证明,:,AA,*,=,A,*,A,=|,A,|,E,.,证明,:,AA,*,=,a,11,a,1,n,a,n,1,a,nn,A,11,A,n,1,A,1,n,A,nn,=,n n,a,1,k,A,1,k,a,1,k,A,nk,k,=1,k,=1,n n,a,1,k,A,1,k,a,1,k,A,nk,k,=1,k,=1,=,|A,|,|,A,|,.,第一章 矩阵,1.6,方阵的行列式,定理,1.9,.,方阵,A,可逆的充分必要条件是,|,A,|,0.,当,|,A,|,0,时,有,A,1,=,|,A,|,1,A,*,.,推论,.,设,A,B,为方阵,若,AB,=,E,(,或,BA,=,E,),则,B,=,A,1,.,事实上,AB,=,E,|,A,|,0,A,可逆,B,=,EB,=(,A,1,A,),B,=,A,1,(,AB,)=,A,1,E,=,A,1,.,A,非奇异,(nonsingular),第一章 矩阵,1.6,方阵的行列式,例,9,.,求下列方阵的逆矩阵,.,(1),A,=,1,2,3 4,1,2,3,2 2 1,3,4 3,(2),B,=,.,解,:(1),A,1,=,|,A,|,1,A,*,=,2,1,4,2,3 1,.,(2)|,B,|=2,0,B,1,=,|,B,|,1,B,*,B,11,=(,1,),1+1,2 1,4 3,=2,B,21,=6,B,31,=,4,B,12,=,3,B,22,=,6,B,32,=5,B,13,=2,B,23,=2,B,33,=,2.,=,2,1,2,6,4,3,6,5,2,2,2,.,第一章 矩阵,1.6,方阵的行列式,例,10,.,设方阵,A,满足,A,2,+3,A,E,=0.,证明,:,A,及,A,2,E,可逆,并求它们的,逆矩阵,.,定理,1.10,.,分块对角矩阵,A,=diag(,A,1,A,2,A,s,),可逆的充分必要条件是,:,A,1,A,2,A,s,都,可逆,.,当,A,1,A,2,A,s,都,可逆,时,A,1,=diag(,A,1,1,A,2,1,A,s,1,).,类似题,P54 34,例,10,（,P47 4,）,已知 ，求,A,-1,2.,克拉默法则,(Cramers Rule),第一章 矩阵,1.6,方阵的行列式,G.,Cramer,瑞士,(,1704.7.31,1752.1.4,),C.,Maclaurin,英,(,1698.2,1746.6.14,),第一章 矩阵,1.6,方阵的行列式,可以表示为,Ax,=,b,.,则线性方程组,x,1,x,2,x,n,记,x,=,b,1,b,2,b,m,b,=,A,=,a,11,a,12,a,1,n,a,21,a,22,a,2,n,a,m,1,a,m,2,a,mn,下面讨论,A,为,n,阶方阵的情形,.,第一章 矩阵,1.6,方阵的行列式,对于,n,元线性方程组,记,D,=,a,11,a,12,a,1,n,a,21,a,22,a,2,n,a,n,1,a,n,2,a,nn,D,1,=,b,1,a,12,a,1,n,b,2,a,22,a,2,n,b,n,a,n,2,a,nn,D,2,=,a,11,b,1,a,1,n,a,21,b,2,a,2,n,a,n,1,b,n,a,nn,D,n,=,.,a,11,a,12,b,1,a,21,a,22,b,2,a,n,1,a,n,2,b,n,第一章 矩阵,1.6,方阵的行列式,定理,1.11,.,设,A,为,n,阶方阵,|,A,|,0,则方程组,有唯一解,:,Ax,=,b,x,1,=,D,1,D,x,2,=,D,2,D,x,n,=,D,n,D,.,证明,:,|,A,|,0,1,D,A,*,b,x,=,A,1,b,=,=,1,D,A,11,A,n,1,A,1,n,A,nn,b,1,b,n,x,1,x,n,总结,(1),行,列式的,6,个性质,(2),计算行列式常用方法：,利用定义,;,利用性质把行列式化为上三角形行列,式，从而算得行列式的值,作业,33,，,第一章 矩阵,1.6,方阵的行列式,例,6,.,证明,n,阶级,(,n,2),范德蒙行列式,D,n,=,1 1 1,a,1,a,2,a,n,a,1,2,a,2,2,a,n,2,a,1,n,-1,a,2,n,-1,a,n,n,-1,=(,a,i,a,j,).,n,i,j,1,Alexandre-Thophile,Vandermonde,Born:,28 Feb 1735 in Paris,France,Died:,1 Jan 1796 in Paris,France,第一章 矩阵,1.6,方阵的行列式,=,1 1 1 1,0,a,2,a,1,a,3,a,1,a,n,a,1,0,a,2,(,a,2,a,1,),a,3,(,a,3,a,1,),a,n,2,(,a,n,a,1,),0,a,2,n,-2,(,a,2,a,1,),a,3,n,-2,(,a,3,a,1,),a,n,n,-2,(,a,n,a,1,),现设等式对于,(,n,1),阶范德蒙行列式成立,则,证明,:,当,n,=2,时,D,2,=(,a,2,a,1,).,D,n,=,1 1 1,a,1,a,2,a,n,a,1,2,a,2,2,a,n,2,a,1,n,-1,a,2,n,-1,a,n,n,-1,(,a,1,),(,a,1,),(,a,1,),第一章 矩阵,1.6,方阵的行列式,=(,a,2,a,1,)(,a,3,a,1,)(,a,n,a,1,),1 1 1,a,2,a,3,a,n,a,2,n,-2,a,3,n,-2,a,n,n,-2,=,1 1 1 1,0,a,2,a,1,a,3,a,1,a,n,a,1,0,a,2,(,a,2,a,1,),a,3,(,a,3,a,1,),a,n,2,(,a,n,a,1,),0,a,2,n,-2,(,a,2,a,1,),a,3,n,-2,(,a,3,a,1,),a,n,n,-2,(,a,n,a,1,),=(,a,2,a,1,)(,a,3,a,1,)(,a,n,a,1,),(,a,i,a,j,),n,i,j,2,=(,a,i,a,j,).,n,i,j,1,