线性代数习题解答(同济大学（第四版）)(2)

100 b?ac?bd?bb2(b?a)c2(c?a)?b2(b?a)d2(d?a)?b2(b?a)=(b?a)(c?a)(d?a)(c?b)(d?b)?

11 2222(c?bc?b)?a(c?b)(d?bd?b)?a(d?b)=(a?b)(a?c)(a?d)(b?c)(b?d)(c?d)(a?b?c?d)

(5) 用数学归纳法证明

x?1当n?2时,D2??x2?a1x?a2,命题成立.

a2x?a1假设对于(n?1)阶行列式命题成立，即

n?1 Dn?1?x?a1xn?2???an?2x?an?1,

则Dn按第1列展开:

?10?00?1?00n?1xDn?xDn?1?an(?1)?xDn?1?an?右边 ?????11?x?1所以，对于n阶行列式命题成立.

6.设n阶行列式D?det(aij),把D上下翻转、或逆时针旋转90?、或依

副对角线翻转，依次得

an1?anna1n?annann?a1nD1???， D2??? ，D3???a11?a1na11?an1an1?a11证明D1证明

，

?D2?(?1)?D?det(aij)

n(n?1)2D,D3?D.

a1nann ?a2na11?an1?annn?1an1??D1????(?1)?a11?a1na21?a11?a1na21?a2nann?? ?(?1)n?1(?1)n?2an1??a31?a3n6

a11?a1n?(?1)n?1(?1)n?2?(?1)??

an1?ann?(?1)1?2???(n?2)?(n?1)D?(?1)D

a?an1n(n?1)11n(n?1)n(n?1)同理可证D2?(?1)2???(?1)2DT?(?1)2D

a1n?ann D3

7.计算下列各行列式（Dk为k阶行列式）：

n(n?1)2?(?1)n(n?1)2D2?(?1)n(n?1)2(?1)n(n?1)2D?(?1)n(n?1)D?D

a(1)Dn1?,其中对角线上元素都是a，未写出的元素都是0；

?1aax?a????aa; ?x?(a?n)n?(a?n)n?1???a?n?1bn?0?;

xa(2)Dn??aan(a?1)nan?1(a?1)n?1(3) Dn?1???aa?111an;

提示：利用范德蒙德行列式的结果．

a1b10(4) D2n?0c1d1?0?cndn(5)Dn?det(aij),其中aij?i?j;

7

1?a1111?a2(6)Dn???11解

?1?1,其中a1a2?an?0.

???1?ana00(1) Dn??010a(?1)n?10?00a0?0000a?00??????000?a0100按最后一行展开

?0a00a?0000?0?????000?a1a0 ?(?1)2n?a?0a(n?1)(n?1)?0(n?1)?(n?1)(再按第一行展开)

a?(?1)n?1?(?1)n?a(n?2)(n?2)(2)将第一行乘(?1)分别加到其余各行，得

?an?an?an?2?an?2(a2?1)

xaaa?xx?a0Dn?a?x0x?a???a?x00再将各列都加到第一列上，得

?a?0?0 ??0x?a?a?0?0 ??0x?a8

x?(n?1)aaa0x?a0Dn?00x?a???000?[x?(n?1)a](x?a)n?1

(3)从第n?1行开始，第n?1行经过n次相邻对换，换到第1行，第n

n(n?1)行经(n?1)次对换换到第2行…，经n?(n?1)???1?次行

2交换，得

Dn?111n(n?1)aa?1?(?1)2??an?1(a?1)n?1an(a?1)nn(n?1)2?1?a?n?? ?(a?n)n?1?(a?n)n此行列式为范德蒙德行列式

Dn?1?(?1)?(?1)?

n(n?1)2n?1?i?j?1?[(a?i?1)?(a?j?1)]

n(n?1)2n?1?i?j?1?[?(i?j)]?(?1)?(?1)n?(n?1)???12?n?1?i?j?1?[(i?j)]

n?1?i?j?1?(i?j)

an?0a1c100?a1b1c1d10??0??dn?100dn

bnb1d1?

(4) D2n?0?cnan?1?dnbn?10按第一行0an展开cn?10?9

0an?1??(?1)2n?10?a1b1c1d1bn?10?dn?10bn0?cn?1cn

0都按最后一行展开andnD2n?2?bncnD2n?2

由此得递推公式： D2n即 D2n?(andn?bncn)D2n?2 ??(aidi?bici)D2

i?2na1b1而 D2??a1d1?b1c1

c1d1得 (5)aijD2n??(aidi?bici)

i?1n?i?j

012310122101Dn?det(aij)?3210????n?1n?2n?3n?4?1111?1?111r1?r2?1?1?11r2?r3,??1?1?1?1????n?1n?2n?3n?4????????????n?1n?2n?3

n?4?0111c2?c1,c3?c1

1c4?c1,??010

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