线性代数(本)习题册行列式-习题详解

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5x12 ??0x?181232155x53 121?3x含x4，x3的项仅有主对角线上元素之积项，故x4，系数分别为15，-3.

三、解答题

1.计算下列行列式 .

1234(1)

23413412；

4123解：各行加到第一行，得

101010101111原式=

23413412?1023413412

41234123 Word 文档

11111111 =10012?101?2?1?10012?100?40?160.

0?3?2?1000?4x3的1111112345(2)122324252； 123334353124344454

解：原式=(5-4)(5-3)(5-2)(5-1)(4-3)(4-2)(4-1)(3-2)(3-1) =288.

14916 (3)

4916259162536；

16253649 .

1491614916原式=

3579357957911?2222?0.

79111322220y0x (4)

x0y00x0y；

y0x0xy0x0y原式=?y00y?x0x0 yx0y0x =y2xyyx?x2xyyx??(x2?y2)2.

1xyz (5)1yzx； 1zxyWord 文档

1xyz原式=0y?x?z (y?x) 0z?x?y(z?x)=?(y?x)(z?x)1z1y?(x?y)(y?z)(z?x).

1010002?100(6)31000；

000210000?2

10100110 原式=?202?1013100??402?1??402000231001 =?42?11?3?20.

1?1?3

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1?x1111(7)

11?x211111?x；

311111?x41?x1?x1?x1?x1解：原式=

1x20010x30

100x41?x1?x1x1x?x?x1?x1?x1?x123x=

04x200 00x30000x4 =

x1x2x3x4?x2x3x4?x1x3x4?x1x2x4?x1x2x3.

1?5132.设D?11341123，计算A41?A42?A43?A44的值.

2234Word 文档

其中A4j(j?1,2,3,4)是D 的代数余子式.

1?513解：A13441?A42?A43?A144?1123?6.

11113?5213. 已知D?110?1?1311,求

M11?M21?M31?M41.

2?4?1?1解：M11?M21?M31?M41

=1?M11?(?1)M21?1?M31?(?1)M41

1?521 =

?110?11311=0.

?1?4?1?14.计算下列n阶行列式.

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21?(1)

11； ?1y解：原式=?x?(n?1)y?y1xy1?y?x?1yy 12???11?2n?11?111?解：原式=

n?12?112????=(n?1)??n?11?211?11?1 =(n?1)01?0????n?1. 00?1xyy?yyxy?y (2)yyx?y ； ????yyy?x Word 文档

11? 2????yyy?x111?10x?y0?0 =?x?(n?1)y?00x?y?0????000?x?y =?x?(n?1)y?(x?y)n?1.

011?11x10?0(3)10x2?0(xi?0,i?1,2,?,n). ????100?xn