线性代数课后习题答案全习题详解

2A2B2???0?1???23????43?? ????3???0?3??0?9??1A1E??EB1??A1A1B1?B2??0?所以 ???OB???OAB???0OA??2??2?22??0??1?0即 ?0?0? 27? 取252?12?4?? 0?43?00?9??210010200??11??01??0?3???0031??112?1???00?23??0?00?3???0252?12?4?? 0?43?00?9??1A?B??C?D???0?0?? 验证AB? |A||B|?

CD|C||D|1?? 解

1AB?0CD?10010?11010021?00?110020?1001000?2010?4? 002011而

|A||B|11 ??0? |C||D|11AB? |A||B|? CD|C||D|故

?34O??4?3? 28? 设A???? 求|A|及A?

20?O22???8

4

解 令34? A??20?? A1???4?3??2?22?????则

A1O?A???O??

A?2?8OAO??8?AA??1???18?? ?OA?2??OA2?8故

888816|A8|?|A1||A2|?|A1||A2|?10?

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?540O?4?O??0544?A1A???? 4???4OA20?2??O64?22?? 29? 设n阶矩阵A及s阶矩阵B都可逆? 求

?OA? (1)??BO???1?

C1C2?OA??? 解 设????CC?? 则 BO???34?

?1OA??C1C2???AC3AC4???EnO?? ??BO??CC??BCBC??OE????34??1s?2???AC3?En?C3?A?1?AC4?O?C4?O由此得 ???

BC1?O?C1?O?BC?E?C?B?1?2s?2?1OAOB??? ?????1所以 ????BO??AO??1 (2)

AO??CB?????1?

?D1D2?? 则 ?AO?? 解 设?????CB??D3D4?

?1AD2??EnO?AO?D1D2???AD1??CB???DD??CD?BDCD?BD???OE??

???34??1324??s??D1?A?1?AD1?En?D2?O?AD2?O由此得 ???

CD1?BD3?O?D3??B?1CA?1?CD?BD?E?D?B?1?24s?4?1AOA??? ?????1?1O所以 ???1?CB?BCAB?????1 30? 求下列矩阵的逆阵?

30

?5?2 (1)?0?0? 解 设210000850?0?? 3?2??2?? B??83?? 则

?52?1????2???1?2?? B?1??8???51????25???1?15A???2?

5A?1???2?3???2?3??

??2????58??1?5?2于是 ?0?0?210000850??1?200??1?10???A???A????2500??

???1??03??02?3?BB?????00?58?2?????1?1 (2)?2?1? 解 设021200310?0?? 0?4??1A???1?0?? B??3?12???0?? C??2?14???1?? 则

2??

?1?1 ?2?1?021200310??1?0??AO???A?1O?

????1?1?0CB???BCAB?1????4?0??0??? 0?1??4??100?1?11?0?2211 ??1???263?15??1??82412

第三章 矩阵的初等变换与线性方程组

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1．把下列矩阵化为行最简形矩阵：

?102?1???(1) ?2031?; (2)

?304?3????02?31???03?43??; ?04?7?1????1?3(3) ??2?3??1?3?2?33534?43???41?; (4)

?20??2?1???23??12?3?2?2?3?1?3?7??0?2?4?.

830?743???102?1?r3?r2??001?3??~?0010????102?1???001?3?? ?0003????1000???0010?? ?0001????102?1?r2?(?2)r1??~解 (1) ?2031??304?3?r3?(?3)r1???102?1?r2?(?1)??00?13~??

?00?20?r3?(?2)??r3?3?102?1?r2?3r3?001?3~???~?0001???

?102?1?r1?(?2)r2??0010~???0001?r1?r3???02?31?r2?2?(?3)r1??~(2) ?03?43?

?04?7?1?r3?(?2)r1???02?31?r3?r2??0013??~?00?1?3?r1?3r2???02010?r1?2??0013??~?0000????0105???0013?? ?0000????1?3(3) ??2?3??1?3?2?33534?43?r?3r?21?41? ~?20?r?2r31??2?1?r4?3r1?1?13?43?r?(?4)??200?48?8??~?00?36?6?r?(?3)?00?510?10?3??r?(?5)402?3??1?22?

000?000???1?1??00?00?00?3111?4?2?2?23??2? 2?2??1?1r1?3r2??00~?r3?r2?00?00r?r?42

?23?12(4) ??3?2?2?3?1?3?7?r?2r2?10?2?4? ~830?r?3r32?743?r4?2r21?r?2r?0?1111??2120?2?4??~?0?88912?r?8r?0?77811?31??r?7r41?0?1??10?00?00?120011??0?2?

14?14?? 32