线性代数第五版答案（全）

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?

(1)???210023?11???304?3??

?解 ???210023?11??(下一步? r2?(?2)r1? r3?(?3)r?304?3?1? )

? ~???0100?21?31??(下一步? r2?(?1)? r3?(?2)? ) ?00?20?

? ~???010021??31??(下一步? r3?r2?0010?? )

? ~???010021??31??(下一步? r3?3? )

?0003?? ~??1?00021??13??(下一步? r2?3r3??0001? )

? ~??10?0021?01??(下一步? r1?(?2)r?0001?2? r1?r3? )

? ~??1000??0010??

?0001??(2)??02?3?03?431???

?04?7?1??解 ??02?03??3431??(下一步? r2?2?(?3)r1? r3?(?2)r1?04?7?1?? )

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~??0?02?00?30?11?31??3?(下一步? r3?r2? r1?3r2? )

? ~??02010??0?000013?0?(下一步? r1?2? )

? ~??0?01005??00103?0??

??1?13?43 (3)??3?35?41??? ?2?23?20?

?3?34?2?1??? 解 ??31??3135??4431??(下一步? r2?3r1? r3?2r1? r4?3r1? )?2?23?20?

?3?34?2?1??? ~??01?01?34?48?38??(下一步? r2?(?4)? r3?(?3) ? r4?(?5)? )?0?000??35106??106????1? ~??00131??4232??(下一步? r1?3r2? r3?r2? r?00?011?4?r2? )

?222?02???1?102?3 ~??001?22???0?

?00000000?0??? (4)?2?13201??32??74???3?2??

?2387340?3??? 解 ?2?13201??32??74??(下一步? r1?2r2? r?3?2833?3r2? r4?2r2? )

?2?3740?3??? ~?0?1?2101?21?41??(下一步? r2?2r1? r3?8r1? r4?7r1??0?88912? )

?0?77811???0 ~??1?012101?21??(下一步? r1?r2? r2?(?1)? r4?00014??r3? )

?00014?? 26

? ~??010?001?21?01??21??(下一步? r2?r3? )

?0000014?0???1020?2? ~??0?01?103??

?00000104?0?? 2? 设??0?11000??A??100110?????142536??? 求A?

?001????001????789?? 解 ??0?11000??是初等矩阵E(1??001? 2)? 其逆矩阵就是其本身?? ??101??0?0100?1?是初等矩阵E(1? 2(1))? 其逆矩阵是

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

? A???010??123??10?1??1?0000??1?456??010?

???789????001???456??10?1??452? ???123??010???12?789????001????782?2??

? 3? 试利用矩阵的初等变换? 求下列方阵的逆矩阵?

(1)??3?321??3125?3??

? 解 ??321100??321100??315010?~?0?14?11?323001????002?100?1?

? ~??3203/20?1/2??3007/2?0?1011?2?121?9?/22?? ?002?101?~?0?10???001?1/201/2?? ~??1007/62/3?3/2?0?00101?1?/21?011/2??2? ???72故逆矩阵为???63?32???11?12??

??01?22?? 27

? (2)?3?0?2202?11????01?21?3?

2?2?1??

? 解 ?3?0?2?1?20?11000?2210100?

?01?32?210000100?1??? ~??01?21?3?20010??0429510100?0301??

?02210100??? ~?1?2?3?20010??0?0012110001?

?00?2?110101?30??4?2??? ~?1?2?3?20010??01210001??0 ?00001112101??36??104????1?200?1?1?2? ~??010001`0?2?1??0010?1?136?

?000121?6?10???100011 ~???00100?01?1?210?3?4?01061??

?000121?6?10???11?故逆矩阵为????01?120??41???

?211?36?106??? 4? (1)设A???4?221?21??? B???21?32??? 求X使AX?B?31?1????3?1??? 解 因为

(A, B)???41?21?3?r?100102??2?321?11 23?2?1?~ ???0?01001 ?1512?3?4??

?所以 X?A?1B???102???15?3??124??

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