0B?BB??1100?10?0??10AA?1?01?01000??0???0???A?1?01?? ??11000??0?1??
?AC????0例18 设A和B都是n阶矩阵, 0???B? ,则C??
?(A)??AA0??0???0BB???(B)??BB???0AA???? ??(C)??AB0??BA0???0BA???(D)????0AB???? ?1C?1???AO??不妨设A,B都可逆
B?1??O??
?1C??CC?1?AB??AO???OB?1???
?1???BAAO??O???OABB?1??BA??????OAB????
2009题
?O设A和B都是2阶矩阵,A?2, B?3?? .则 ?B?(A)??O3B?2B????2A?O???O?(B)???3A?O??? (C)??O3A??2A????2B?O???O?(D)???3B?O??? ( 2009年的考题)
??1解:
C?CC
?1先求C
A??O???()?
?CE????O??B???1A0O00010000100????0??B??O0?????1??0O0A10000110000??1?0??0??
?E???O?OOEA?1B??O??
?1C??OBA?O??1O??AB??O??????1O??A?1B???O?
ABB??O?????BA?O????1?1?O?AB??A?1??1OB??????1?BAAO???AB??O??
?例16 设A是n阶非零实矩阵,满足 A?A. 证明:
T(1)A?0
(2)如果n?2则A?1
解：条件A即
??A,即
T(Aij)?(aij),TT
Aij?aij,?i,j（1）
A?ai1Ai1?ai2Ai2??ainAin22
2?ai1?ai2???ain又因为 A?0, 即
2?0
A有非零元素，
22则
A?ak1??ake???ain?0AAT
(2)
?AA?AE?
A2?An 得
An?2?1 因为
A?0
n?2是正整数，得A?1
例17 设矩阵
A?(aij)3?3A?A, a11,a12,a13 为3个相等的正数,则它们为 满足
?T(A)设则
33 (B)3(C)13(D)3(2005年数学三)
a11?a12?a13?a,
A?A?A?3a?0?T2
A?1又n?3?2.得 ?3a?1,a?例8 3维向量
2213,a?33.
?1,?2,?3,?1,?2,?3满足
?1??3?2?1??2?0,
3?1??2??1??3?0 ??2??3??2??3?0
已知解：
?1,?2,?3?a,求
?1,?2,?3.
?1??3??2?1??2
3?1??2???1??3 ??2??3??2??3
(?1??3,3?1??2,??2??3)?(?2?1??2,??1??3,?2??3)
?1?(?1,?2,?3)?0?0?13?10
23?1000???2???1??(?1,?2,?3)?1?01????2?101?10?101?1
0??1??1??
?1,?2,?301?1??1,?2,?3110?4a??1,?2,?3P?(?,A?,A?)可逆,并且
例9设A 是3阶矩阵,?是3维列向量,使得
A??3A??2A?.又3阶矩阵B满足A?PBP(1)求B.(2)求解：
32?1A=PBP-1.
A?E?1.(01一)
A?PBP2即AP?PB
22A(?,A?,A?)?(A?,A?,3A??2A?)
?00??(?,A?,A2?)?0?103????01?2??
?00?B??0?103????01?2??
则
A?E?PB?EP?1
100?B?E?113??401?1
?A??2?1例10 3阶矩阵A,B满足ABA??2BA??E?,其中
?0B.(04一)
解：ABA??2BA??E(A?2E)BA??E
A(A?2E)B?A
111A3A?2EB?AB?A?2EA2?1?32?9
?3?51?A???1?10???例11 设3阶矩阵,
??102?? A?1XA?XA?2A,求X.
(A?1?1?XAA?1?2AA?1解：
XA)A
A?1X?X?2EX?AX?2A
(E?A)X?2A
10?20??01??,求
??2?(E?A2A)???1?1???1???0?0?21202520?2?6?21062?10?20210?1?22??0?4??
02?24100?21?40??2?0??
?12?100???1??2???0?04????1???0?0?0100?60?21?4104104??2?0??
??6?X???2??4?得
104104??2?0??
11?1?1??1??1?? AX?1?A???1?1?例12 设3阶矩阵,
?A?1?2X,求X.
解：
AX?A??1?2X
AX?E?2AX(4E?2A)X?E?1X?(4E?2A)
1A??1
11?1?1211?1?1?11?41?11?0101
例13 4阶矩阵
A,B满足ABA?BA?3E,已知
?1??0?A??1??0?
解： ABA?1010?3?BA0010?10??0?0??8??求B. (00一) ?3E
