??210?故 A?1?1??131?AA????3??
?22???167?1??初等行变换法：
??12?1100??AE???34?2010??5?41001? ???r?12?1100???r2?3r13?5r1????0?21?310??0?146?501?
????12?11???r3?7r2??00??0?21?310??
??00?116?71??r?120?157?1????r2??rr313???0?2013?61??0?116?71?
?0???100?210????r3?r2???0?2013?61???00?116?71?
???10?210?????12r2?0????010?131??00?123?2??
16?71?????210?所以 A?1??13?1??3??．
?22???167?1????1?a11（４）由对角矩阵的性质知 A?1???a2???0??0????. ?1?a?n???210. (1) X???15??3??1?4?6? ??21???3?5??4?6??2?23????????? ??12??21??08??21?1?1?13????210(2) X????? 432?????1?11??101?1?13??1?????23?2?? 3?432????330????1??221?? ??82?5?????3??3?14??31??20?(3) X???????
??12??0?1???11??1?1?1?2?4??31??10??????? 12?11??0?1??12??11?1?66??10??? ???????10?12?30??12??4?211. 由A?A?2E?O得A?A?2E 2两端同时取行列式: A?A?2
2即 AA?E?2,故 A?0 所以A可逆,而A?2E?A
2A?2E?A2?A?0 故A?2E也可逆.
由A?A?2E?O得
22A(A?E)?2E
所以 AA(A?E)?2AE，则A?1?1?1?1(A?E) 2又由A?A?2E?O(A?2E)A?3(A?2E)??4E
2(A?2E)(A?3E)??4E
所以 (A?2E)?1(A?2E)(A?3E)??4(A?2E)?1
则 (A?2E)?12.?E?A?13. 因为A?1?114(E3?. A)?E?A???Ak?1.
?1?1?A，所以 A1?11A?5AA?1?A?1?5A?1 22?2A??1?5A??3??2A?1???2?A?1??8A?1??8?2??16.
14. 由A?1?1?A，得A??AA?1， A?n所以 当A可逆时，有A?A从而A也可逆.
因为A?AA，所以
??1?
A?1?An?1?0，
?A?又A???1?A?1A
1?1??1?，所以 A?AA????1?A?A???A?12?1A?A?1A?A?1???A?1?
??15. 由AB?E?A?B得
?A?E?B?A2?E
即
?A?E?B??A?E??A?E?
00110??1?0，所以?A?E?可逆，则
100因为 A?E?0?201???B?A?E??030?.
?102????600?
??16.?020?.
?001????033???17.??123?
?110???18. 因为AP?P?，所以A?P?P；
?1??100??1????5??11又 P??1， P???210?，?=?? ??41?1?55??????100???1???100????????11???210? 所以 A?P?P??2?10???211???5???????41?1??100?????200? ?6?1?1????100???A5??200?.
?6?1?1????1?1?1?119. 因为A?BA?E?CA?BB?BA，由?A?B??A?B????1?E得
?A则
?1?B?1?A?A?B??B?1?A?B??A?B??B?1
?1?1?1?A?B?1?A?A?B?B?B?1B?E
?1?1?1?1所以A?B可逆，其逆为A?B?A?B.
?2?131?r?2r?2?131???r32?r21??4?25400?12r4?r1????? 20. A??????42?6?2??00?12?????2?140001?1?????2?13?r3?r200?1r4?r2??????000??0001??2?13??2?r3?r4?00?1??????0000??1??000?1311??2??B ?1?0?B的秩为3，其一个3阶非零子式为00?1?223511?2，对应于A的3阶非零子式为014. ?6?2?2?1?00故??00??00?1?223514. ?6?231??1?2?即为矩阵A的行阶梯形矩阵，矩阵A的一个最高阶非零子式为
01??00??1??2?121.(1)??4?5???4121?41?41??72??6??1?,(2)??14??1?3??????24?23???32??12?，
1?0?2??11?2?4??1?311?20?????010?101?21?，(4)??. (3)???1?13?001?2?6?????21?6?100001????22.(1)2,(2)3，(3)4，(4)当a??4时，秩为2；当a??4时，秩为3.
?34?O??4?3?，令A??34? A??20? 24.A??21?????224?320??????O?22??则A???A1?OO?? A2?