线性代数习题解答(同济大学（第四版）)(3)

?100?1?20?1?2?2?1?2?2???n?12n?32n?41?a1111?a2(6)Dn???11a100??a2a20?0?a3a3?00?a4?????000?000?0?00?00?0n?1n?2=(?1)(n?1)2

?2?0???2n?5?n?1?1?1c1?c2,c2?c3

??c3?c4,??1?an001001001按最后一列001

展开（由下往上）????an?1an?110?an1?ana100?000?a2a20?0000?a3a3?0000?a4?000 (1?an)(a1a2?an?1)?0???????000??an?2an?20000?00?ana100?00?a2a20?000?a3a3?00????

??????000??an?1an?1000?0?an11

?a2a20?000?a3a3?0000?a4?00

??????000??an?1an?1000?0?an?(1?an)(a1a2?an?1)?a1a2?an?3an?2an???a2a3?an

n1?(a1a2?an)(1??)

i?1ai

8.用克莱姆法则解下列方程组：

?x1?x2?x3?x4?5,?x?2x?x?4x??2,?234(1)?1

?2x1?3x2?x3?5x4??2,??3x1?x2?2x3?11x4?0;?1,?5x1?6x2?x?5x?6x?0,123??(2)?x2?5x3?6x4?0,

?x3?5x4?6x5?0,??x4?5x5?1.?1111111112?1401?23(1)D? ?2?3?1?50?5?3?7312110?2?1810?00111?20?130?51130?801401111?23??142

0?1?540014251110509

?2?3?1?501211解

5111?22?14D1???2?3?1?50121112

1?5?1?90509??0?13?3?23012111?5?1?901211

05090?13?3?231?5?1?901211??00?10?4600231201?5?1?901211??142

00?138000142151115111?2?140?7?23 D2??2?2?1?50?12?3?7302110?15?18151115110?1320?132????284 00231100?1?19003931000?284115112?24D3???426

2?3?2?531011111512?1?2?142 D4?2?3?1?23120?D3D1D2D4x1??1,x2??2,x3??3,x4???1DDDD6510006510006510501按最后一行?5D?00展开60565100651

51(2)D?00000?5D??6D?? 06?5(5D???6D???)?6D???19D???30D???

13

?65D????114D?????65?19?114?5?665

?的余子式(D?为行列式D中a11的余子式,D??为D?中a11,D???,D????类推) 10D1?00151D2?0006510010001065100651000651006510605按第一列?D?01展开6050160按第二列05?001展开60050651006500 06?D??64?19D????30?????64?1507

0651050016?605501006500 06560?156?5?63??65?1080??1145 01551D3?0006510010001006510150按第三列010展开0060050651056015?601500006500 06160560?056?6150?19?6?114?703 01501651D4?0006510006510100010150按第四列01?000展开600565100501?60506510065100 06560??5?6156??395

01514

51D5?000?651000651000651110按最后一列000展开0015100651006?D??1?211?212 51x1?15071145703?395212;x2??;x3?;x4?;x4?． 665665665665665??x1?x2?x3?0?9.问?,?取何值时,齐次线性方程组?x1??x2?x3?0有非零解？

?x?2?x?x?023?11解 D3?1?1?????，

12?1齐次线性方程组有非零解，则D3即 ?得 ?不难验证，当?

?1?0

????0

?0或??1

?0或??1时,该齐次线性方程组确有非零解.

?(1??)x1?2x2?4x3?0?,齐次线性方程组10.问?取何值时?2x1?(3??)x2?x3?0

?x?x?(1??)x?023?1有非零解？ 解

1???241???3??4D?23??1?21??1

111??101???(1??)3?(??3)?4(1??)?2(1??)(?3??) ?(1??)3?2(1??)2???3

齐次线性方程组有非零解，则D?得 ? 不难验证，当?0

?0,??2或??3

?0,??2或??3时，该齐次线性方程组确有非零解.

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