线性代数习题册行列式-习题详解

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

412341231111111 =10012?11?2?1?10012000?40?3?2?10001?10?160.?41111112345(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；

162536491491614916原式=

3579357957911?2222?0.

7911132222

0y0x (4)

x0y00x0y；

y0x0xy0x0y原式=?y00y?x0x0 yx0y0x =y2xy2xy222yx?xyx??(x?y).

1xyz (5)1yzx； 1zxy1xyz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

101010 原式=?202?103100??402000231 =?42?11?3?20. 1?x1111(7)

11?x211111?x31；1111?x411?1??4000012?11?3

1?x1?x1?x1?x1解：原式=

1x20010x

30100x41?x1x1x11?xx???x1?x12x3x?x1=

04x200 00x30000x4 =

x1x2x3x4?x2x3x4?x1x3x4?x1x2x4?x1x2x3.

1?5132.设D?11341123，计算A41?A42?A43?A44的值.2234其中A4j(j?1,2,3,4)是D 的代数余子式.

1?513解：A113441?A42?A43?A44?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.

21?1(1)

12?1???； 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?x11? 2111?1yxy?y解：原式=?x?(n?1)y?yyx?y ????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