?2k?000?假设Ak??02k00????a02k0?，则 ?k?0ak02k?????2k000?000??2k?10Ak?1??02k00??2????0200?2k?1?ak02k0??1020?????0?2a?k?0?0ak02k?????0102???2k??02ak?2kaak?1ak1aakkk?1?2ak?2k?2k?1?2k?2，且12?12?2k?2 ?a1k?k?2k?
?2k?000??Ak??02k00???k?1?k202k0? ??0k2k?102k???9．求下列矩阵的秩
?3102???1011?（1）??1101??1?12?1??； （2）??13?44????0111?? ?11-20????1230?1221??2?11?5????1（3）?215－1???10?12? （0?0111?4）??03－13?????2?3?12?7???1104－1??
??解：（1）?3102?r2?r1?1?12?1??1?12?1??1??3102?r?3r??12104?13?44?????13?44???r3?r?1??04??1?12?1??04?65?? ?R?A??2 ??0000??35
00?00??2k?10? 02k?1???2?1??65??65? ???1?1（2）??0??10110111?21??1?1??0??1??0?0??0011??1?10?111??1?3?1??1??0?0??0011??1?10?021??0?2?1??1??0?0??0011?102001??0? ?1?0??R?A??3
??1230?30??1230??2?11?5????12?0?5?5?5????0111?（3）???10?12??0222??0000??
?0111????0111????0000????3?12?7????0?7?7?7????0000???R?A??2
??11221??112221?（4）?0215?1?5?15?1???13??1???021???112?20?0?2?1?51???0213?00?2?2?? ?1104?1????00?22?2???2?00000???R?A??3
10．求下列矩阵的秩及行的最简形
??1?2?102??3?20?（1）??2426?6???2?1023??1???； （2）?0221?. ?33334????1?2?3?2??0121????1?2?102??1?2?102??1?2?10解：（1）??2426?6?6?2?????000?2?1023?1???03222???000?33334????032?0963?2???6?000?3??1?2?102??2????1?2?10??101160122?30?9??3?1?3???0?1?3?0129???21??01??00?1??3??30?9?
?01?0001?1?3???3???01?1???00000???00000???00?3???00000???R?A??3
36
2??1??2??1??
???1?2?3?2??3?20?1???1?2?3?2???0111??（2）?0221??r3?r1??0221???2??1?2?3?2?r3?3r1?0495??0053? ?0121????0121?????0011??2????1?2?3?2???0111??1?2?30??00????2??010??1?20100??10010??0011??12??0????1???001001???0010???0??0001???00?000??0001??02???R?A??4
11．求下列方阵的逆
?12?1?（1）??34?2?（2）?cos??sin????； ?cos??； ?5?41???sin????1000?200?（3）?1200??100??0??5?； （4）?2?213?04?2??； ?1214?0???0013???0013?
?3?20?1??0
0028?
（5）?0221??
???0； 0100??.
?1?2?3?2? （6）?1
?0121??
??
23200?
???3
1100??
?12?1100?r?解：（1）?2?3r1?34?2010??12?1100????0?21?310? ?5?41001??r3?5r1??0?146?501???r?3?7r?12?1100???12?0?21?310??120?157?1??0?2013?61??0??00?116?71???0?2???00?116?71?????0037
0?0?0?? 1??0?21013?6?116?70?1?1????100?210???210?????131131?1?010?3?? ?A???3??
?22??22??001?167?1???167?1?????cos?（2）A??sin?sin??1，A11?cos?，A12??sin?，A21?sin?，A22?cos? cos?A??A11?A???A?A12?1A21??cos????A22???sin?1000010000100??0?0??1??1??0?0??0sin??? cos??02120031010?10?24?1010000100??0? 0??1??1?（3）?1?2??1021200310004?1??0??0??0??1??0???0???0?0002001?1320030?014000?100???000????10?10???2???101???0?010121?65?2400002001?13030?210042011?25?60??0??10?
?1??1?3?0011100?21010?2100180??00?2400????121201?1?? ?A?10?24??12?483???3?5?2?1?1?12??0??0?0?6?
?5?2?（4）A??0??0?112100?100??00??A1?4?2??O?13?O?? A2??1?52??1?2??4?2?1?32??1A???A?????，2????
21?2513?1414????????38
?AA?1??1?OO??A1?1???A2??O?10?1?2?0??25O??3??00?1?A214??1?00??14?010000100??0?1? ?7?2??7?001001100?3000??0? 0??1??3?20?1?（5）?0221?1?2?3?2??012110000??1?2?3?2??0?r3?r1?02210?r3?3r1?0495??1??0121?1?2?3?2??0111?2?0053?1?001??201021?210?200??00???30???01??1?1?2?3?2??0111?2?1?0012??1?000?2010210?211200??00???
01????3?5??1?1?2?3??011?001??000?1??0?0??001000010042?11?20??0?1035?0?1?136??121?6?10??1?2??01?00??00001001?1?2?2??0010?1? ?0?1?136?121?6?10?4??1? 6??1?0011?2?4?1?2??1??0010?1?010? ?A?1????1?130?1?136???121?6?10?1?6??20013??0028??O?0100???A23200?1100???0??0（6）A??1??2?3?A1?? O??1??6?13???101???14????13???2?1??2A?232?A1?1???，??2????3?28???11???????311?7?2????6?1613161?2??0? ??1???2?39