方阵的逆-矩阵方程-wlk

1第十二讲方阵的逆方阵的逆 矩阵方程矩阵方程2例43:).det()det(,n,mBAIABImBnAnm证明矩阵为矩阵为设解:,00,00nmnmnmnmnmnmIAABIIBIIBAIIBAAIIBAIIBI:det()det().mnIABIBA两边取行列式得也成立号对3011010010101nnnTTnnnTTIIIIII 001110001101nnnTTTTnnnTTIIIIII11:()(1)TTTnnII 证明411111111001010(1)10(1)10(1)(1)(1)(1)(1)nnnnTTTTnnTTTTTnTTTIIIIIII 110()0110101TnnnnTTIIII11()(1)TTTnnII5101111110 11T记Example1:计算解:1110111()(1)1110211111()111112TTTTIInInnn 6AXB解 矩 阵 方 程46115:6911ExampleX解矩阵方程.,119664:矩阵方程亦无解无解方程组解X12(2),.nAAXBAXBAXBAXB若不可逆 则有解有解211(1),.(,)(,)AXA BA BI A B若可逆 则7143132111830520002X解矩阵方程Example6:,110100341010001000110001143830132520111002212121212121212325212121解:.110341212121X83法.1111111121244I3111131111311113111111112detI.483161111111121214I9Example7:.,A,032A:2并求其逆可逆证明阶方阵且满足是设IAnA证:).2(31,)2(31,3)2(,03212IAAAIIAAIIAAIAA可逆且于是有思考题:.并求其逆,逆IA证明,AA阶方阵且满足n是A设2可,02 AA(A+I)(A-2I)=-2I.22)(1IAIA10推论5:.,单位方阵可仅用初等行变换化为则阶可逆方阵是设AnA10001,1112nnnaaaA用三种初等行变换然后再从最后一行开始往上打洞.,11nssIAPPPP 使一系列初等方阵即.111nssIPPPPA11Example5:.,A,032A:2并求其逆可逆证明阶方阵且满足是设IAnA证:).2(31,)2(31,3)2(,03212IAAAIIAAIIAAIAA可逆且于是有思考题:.并求其逆,逆IA证明,AA阶方阵且满足n是A设2可,02 AA(A+I)(A-2I)=-2I.22)(1IAIA12Example6:.B,71,41,31:,试求满足设三阶方阵diagBA714131,61ABAABAA且解:IBIAAABAIA)(61,6)(111得右乘.123)743(6)(61(111IIAB。