?5?2 (1)?0?0?210000850?0?? 3?2??5 解 设A???2?5 A?1???2?2?? B??8?51????13?? 则 2??3???2?3??

??58?2?????12???1?2?? B?1??8??25??51??????1?5?2于是 ?0?0??1?1 (2)?2?1?02122100003100850??1?200??1?10???A???A????2500??

???1??03??02?3B?B???????2?00?58??0?0?? 0?4??1 解 设A???1?

0?? B??30?? C??21?? 则

?14??12?2???????1?1?1 ?2?1?021200310??10???AO???A?1O?

????1?1?0CB???BCAB?1????4?0??0??? 0?1??4?00?1?11?0?22?1 ??1?1?263?15??1??82412

第三章 矩阵的初等变换与线性方程组

1? 把下列矩阵化为行最简形矩阵?

?102?1? (1)?2031??

?304?3????102?1? 解 ?2031?(下一步? r2?(?2)r1? r3?(?3)r1? )

?304?3????102?1? ~?00?13?(下一步? r2?(?1)? r3?(?2)? )

?00?20????102?1? ~?001?3?(下一步? r3?r2? )

?0010????102?1? ~?001?3?(下一步? r3?3? )

?0003????102?1? ~?001?3?(下一步? r2?3r3? )

?0001????102?1? ~?0010?(下一步? r1?(?2)r2? r1?r3? )

?0001????1000? ~?0010??

?0001????02?31? (2)?03?43??

?04?7?1????02?31? 解 ?03?43?(下一步? r2?2?(?3)r1? r3?(?2)r1? )

?04?7?1????02?31? ~?0013?(下一步? r3?r2? r1?3r2? )

?00?1?3????02010? ~?0013?(下一步? r1?2? )

?0000????0105? ~?0013??

?0000????1?13?43??3?35?41? (3)??

2?23?20??3?34?2?1????1?3 解 ?2?3??1?0 ~?0?0??1?0 ~?0?0??1?3?2?33534?4?4?2?23?1?(下一步? r?3r? r?2r? r?3r? )

213141

0??1???13?43?0?48?8?(下一步? r?(?4)? r?(?3) ? r?(?5)? )

234

0?36?6?0?510?10???10003111?4?2?2?23?2?(下一步? r?3r? r?r? r?r? )

123242

2?2???1?0 ~?0?0??100002?3?1?22?? 000?000???23?12 (4)?3?2?2?3?1?3?7?0?2?4?? 830?743??1?3?7?0?2?4?(下一步? r?2r? r?3r? r?2r? )

123242

830?743???23?12 解 ?3?2?2?3??0?1111??120?2?4? ~?(下一步? r2?2r1? r3?8r1? r4?7r1? )

0?88912??0?77811????0?1 ~?0?0??1?0 ~?0?0??1?0 ~?0?0??100001000100120011?0?2?(下一步? r?r? r?(?1)? r?r? )

12243

14?14??0?2??1?1?(下一步? r?r? )

23

14?00??0010?2?3?? 4?0??2?1002?100?010??101??123? 2? 设?100?A?010???456?? 求A?

?001??001??789????????010? 解 ?100?是初等矩阵E(1? 2)? 其逆矩阵就是其本身?

?001????101? ?010?是初等矩阵E(1? 2(1))? 其逆矩阵是

?001????10?1? E(1? 2(?1)) ??010??

?001????010??123??10?1? A??100??456??010?

?001??789??001????????456??10?1??452? ??123??010???122??

?789??001??782??????? 3? 试利用矩阵的初等变换? 求下列方阵的逆矩阵?

?321? (1)?315??

?323????321100??321100? 解 ?315010?~?0?14?110?

?323001??002?101??????3203/20?1/2??3007/22?9/2? ~?0?1011?2?~?0?1011?2?

?002?10??001?1/201/2?1?????1007/62/3?3/2? ~?010?1?12?

?001?1/201/2???