线性方程组(克莱姆法则)

第二章线性方程组线性方程组的一般形式为本章讨论1)解的存在性2)解的求法3)解的个数4)解的结构何时无解？)(怎样求解？)(解与解之间的关系)11121121222212121212mmnnnnnnmmaaabaaabxxxxxxxxxaaab(有多少个解？)(何时有解？方程组的求解问题:如果存在个数n当方程组的个等式m则称nnxxccxc1212,,...,为该方程组的一个解．方程组的全体解构成的集合，称为方程组的解集.都成立,11,xc22,xc...,nnxc对于方程组基本概念：11121121222212121212mmnnnnnnmmaaabaaabxxxxxxxxxaaab1c2cnc1c2cnc1c2cnc使得时，12,,...,,nccc设有两个n(Ⅰ)的每个解如果方程组(Ⅰ)都是方程组(Ⅱ)的解;同时都是方程组(Ⅰ)的解,则称这两个方程组的每个解,同解.方程组(Ⅱ)11121121222212121212kknnnnnnkkcccdcccdxxxxxxxxxcccd元线性方程组11121121222212121212mmnnnnnnmmaaabaaabxxxxxxxxxaaab(Ⅱ)与§２.１线性方程组首先讨论：未知量的个数方程的个数的方程组.方程组有唯一解：aa1122aa1122a11当x1x2即当≠0时22111()axa22112()axa0时，1221aa12xx1122aa21122211aaaa21122211aaaa11112222aaaa11122122121122xxbxxbaaaa11122122121122xxbxxbaaaa212ba122ba121ba211ba122ba212ba121ba211ba211222bbaa111122aabb11122122aaaa11122122aaaa12a21a22a一、克莱姆(Cramer)法则二元线性方程组11122121122221xxxabaaabx当≠0时,方程组有唯一解：1x1222aadetAdetA1detB2x2detB1121aa这一结果可以推广到一般的含有n个未知量n个方程的线性方程组.detA12bb12bb1112aa2122aa11122122aaaa11122122aaaa三元线性方程组1231111213aaaxbxx当时,方程组有唯一解：detA111213aaa212223aaa1213333233abaxaxx313233aaa1x2x123bbb111213212223313233aaaaaaaaa121322233233aaaaaa0111213212223313233aaaaaaaaa3x111213212223313233aaaaaaaaa111321233133aaaaaa123bbb111221223132aaaaaa123bbb1212232223abaxaxx四元线性方程组1234111213141aaaaxxxbx当时,方程组有唯一解：detA11121314aaaa21222324aaaa2122112232344xxxxbaaaa3132312333344xxxxbaaaa31323334aaaa04142412434344xxxxbaaaa41424344aaaa1x2x1234bbbb11121314212223243132333441424344aaaaaaaaaaaaaaaa3x1234111213141aaaaxxxbx2122112232344xxxxbaaaa3132312333344xxxxbaaaa4142412434344xxxxbaaaa121314222324323334424344aaaaaaaaaaaa11121314212223243132333441424344aaaaaaaaaaaaaaaa1234bbbb111214212224313234414244aaaaaaaaaaaa1234bbbb111314212324313334414344aaaaaaaaaaaa4x111213212223313233414243aaaaaaaaaaaa1234bbbb其中定理2.1(克莱姆法则)12121211121212221212.........nnnnnnnnnnaaaaaaxxxxxxxxaxaabbb当其系数行列式对应后得到的行列式.有且仅有唯一解11detdetBxA22detdetBxAdetdetnnBxA是将系数行列式detA12,,...,nbbb线性方程组21(.)≠0时,11121...naaa21222...naaadetA12...nnnnaaa1detB12(,,...,)jn...地换为方程组的常数项中第1列元素22jjnnnaaaaBaa1112212212......det...有且仅有唯一解：12121211121212221212.........nnnnnnnnnnaaaaaaxxxbbbxxxxxxaaa当时,det0A11detdetBxA22detdetBxAdetdetnnBxA...nbbb12nbbb12...nbbb12两个条件:三个结论:11121...naaa21222...naaadetA12...nnnnaaa1detB1212222.........nnnnnaaaaaa2detB111312123213.........nnnnnnaaaaaaaaa证将方程组表为矩阵形式即12121211121212221212.........nnnnnnnnnnaaaaaaxxxbbbxxxxxxaaa(2.1)AXBbnb12nxxx11121...naaa21222...naaa12...nnnnaaa1bAXBA是n阶方阵.由于故Ａ可逆，得由因此,且解必为从而解存在唯一.12121211121212221212.........nnnnnnnnnnaaaaaaxxxbbbxxxxxxaaa(2.1)AXBAdetAXBA1A1XAB1111212122212.........nnnnnnaaaaaaaaa12nxxxnbbb1A存在XAB1有解,A方程组(2.1)XAB1是方程组(2.1)的唯一解..........nnnnnnnnnnaaaaaaxxxxxxxaaabbbxxnxxxnbbb11121...naaa21222...naaa12...nnnnaaa当时,det0A方程组(2.1)有唯一解X即12nxxxnAAA11211nAAA22221.........nnnnAAA21nbbb1detA1211211...nnAbbAAb1212222...nnAbbAAb1212...nnnnnAbbAAb1detA证毕B1detB2detnBdet即1x2xnx...1ABBA1detdetBA2detdetnBAdetdet1Annnnnaaaaaa1212222.........nbbb12nnnnnnaaaaaaaaa111312123213.........nbbb12nnaaaaaa1112212212.........nbbb12221bA...1nnbA1detB111bA2detB121bA222bA...2nnbAdetnB11nbA22nbA...nnnbA111212122212.........nnnnnnaaaaaaaaa1231231231234429xxxxxxxxx例方程组有唯一解.123111149124213132()()()223232()()()1112349D20(21)(31)(32)D21113192124(21)(21)(22)3111214D12124方程组的唯一解为：1x303x302x3020212D解300常数项均为零的方程(２.１)所对应的111212122212121212.........nnnnnnnnnaaxxxxxxxxaaaaxaaa当然是方程(２.４)的解称为齐次线性方程组(２.４)的齐次线性方程组除零解外,齐次线性方程组..........nnnnnnnnnnaaaaaaxxxxxxxaaabbbxx10,x是否还有其它解?021(.)20,x...,0nx0000000024(.)的齐次线性方程组为:线性方程组称为000零解.1231231232030220xxxxxxxxx例齐次线性方程组是其零解.除零解外,也是其解,例齐次线性方程组xxxx其解必满足此方程组称为非零解12300,0,xxx120x10,x20x只有零解.0000000001,5x24,x33x定理２.２121211121212221122....000.....nnnnnnnnnxxxxxxaaaaaaaaaxxx的系数行列式111212122212.........nnnnnnaaaaaaaaa则它仅有零解.如果含有个方程的n元齐次线性n24(.)detA0方程组证即方程组只有零解.由克莱姆法则,方程组有唯一解．121211121212221122....000.....nnnnnnnnnxxxxxxaaaaaaaaaxxx时,det0A是方程组(２.４)的解，10,x20,x...,0nx24(.)且方程组只有一个解，故10,x20,x...,0nx是方程组(２.４)的唯一解，000000000111122121122221122.........000nnnnnnnnnaxaxaxaxaxaxaxaxax11121...naaa21222...naaadetA12...nnnnaaa方程组只有零解方程组有非零解Adet0例设齐次线性方程组2123123123(1)20(21)20(21)0xkxxxkxxkxkxkx有非零解，求的值．k解detA2112k1212k21kkk2112k2020kk03k122kk(2)kk00k或2k方程组有非零解Adet0(2)kk作业第二版:P111