A?A A*(A*)?1?|A|E(A*)?1?O 所以A*?O
(2)由于A?1?1A*? 则AA*?|A|E
|A| |A||A*|?|A|n 若|A|?0 若|A|?0
有|A*|?0
这与|A*|?0矛盾，故当|A|?0时
取行列式得到
则|A*|?|A|n?1 由(1)知|A*|?0
此时命题也成立
因此|A*|?|A|n?1
?033? 19? 设A??110?? AB?A?2B
??123????1 求B?
故
解 由AB?A?2E可得(A?2E)B?A
?1??233??033??033? B?(A?2E)A??1?10??110????123???121???123??110??????? 20
?101? 设A??020??101??? 且AB?E?A2?B 求B
解 由AB?E?A2?B得 (A?E)B?A2?E
所以(A?E)可逆
从而
即 (A?E)B?(A?E)(A?E)
001 因为|A?E|?010??1?0100?201? B?A?E??030??102??? 21
设A?diag(1
1) A*BA?2BA?8E
求B
?2
解 由A*BA?2BA?8E得 (A*?2E)BA??8E B??8(A*?2E)?1A?1 ??8[A(A*?2E)]?1 ??8(AA*?2A)?1 ??8(|A|E?2A)?1 ??8(?2E?2A)?1 ?4(E?A)?1 ?4[diag(2
?1
2)]?1
?4diag(1, ?1, 1) 22 ?2diag(1 22
?2
1)
?1?0 已知矩阵A的伴随阵A*??1?0? 求B
得|A|?2
010?300100?0?0?8??
且ABA?1?BA?1?3E
解 由|A*|?|A|3?8
由ABA?1?BA?1?3E得 AB?B?3A
B?3(A?E)?1A?3[A(E?A?1)]?1A ?3(E?1A*)?1?6(2E?A*)?1
2?1?0 ?6??1?0?010300100??600???060??60??6???03?100600?0?0??1??
?1?4?? ????10?? 求A11? 23? 设P?1AP??? 其中P???11??02????? 解 由P?1AP??? 得A?P?P?1 |P|?3
所以A11? A=P?11P?1.
1 P*????1?114?1??14 P?1?1???1?1??3??
?1而 ?11???0?0????10 ??0211?2?????14????27312732??1?4?10????1133故 A????0211??11????683?684?11???????????33? 24求
设AP?P?
?111? 其中P??10?2??1?11???
??1??1? ????5??
(A)?A8(5E?6A?A2)
解 (?)??8(5E?6???2)
125)] 1
58)diag(12
0
0)?12diag(1
0
0)
58)[diag(5
5
5)?diag(?6
6
?diag(130)?diag(1
1
?diag(1
(A)?P(?)P?1
?1P?(?)P* |P|?111??100???2?2?2?
??2?10?2??000???303?
?1?11??000???12?1????????111? ?4?111?
?111??? 25
设矩阵A、B及A?B都可逆
证明A?1?B?1也可逆
并求其逆阵 证明 因为
A?1(A?B)B?1?B?1?A?1?A?1?B?1而A?1(A?B)B?1是三个可逆矩阵的乘积 即A?1?B?1可逆
所以A?1(A?B)B?1可逆
(A?1?B?1)?1?[A?1(A?B)B?1]?1?B(A?B)?1A 26
?1?0 计算?0?0?210010200??11??01??0?3???0031?12?1?0?23?00?3??
12? 解 设A1???01????23?B2???0?3???
21? A2???03???31? B1???2?1???
A1E??EB1??A1A1B1?B2?则 ??OA??OB???OAB???2??2?22?
1而 A1B1?B2???0?2 A2B2???0?2??31????23???52??2?1??0?3??2?4?1????????
1???23????43??0?3??0?9?3???????1A1E??EB1??A1A1B1?B2??0?所以 ???OB???OAB???0OA??2??2?22??0??1?0即 ?0?0?210010200??11??01??0?3???0031??112?1???00?23??0?00?3???0252?12?4?0?43?00?9??
252?12?4?0?43?00?9??1 27? 取A?B??C?D???0?1 解 AB?0CD?10010?110100?? 验证AB? |A||B|CD|C||D|1??020?1001000?2010?4002011
021?00?110
|A||B|11而 ??0|C||D|11|A||B|故 AB? CD|C||D|?34O??4?3? 28? 设A??? 求|A8|及A4?20O?22???
34 解 令A1???4?3????A1O?则 A???OA??2?2 A2???2?
0?2??
工程数学线性代数(同济大学第六版)课后习题答案(全)
