a22a?12b?12c?12d?122224a?44b?44c?44d?46a?96b?96c?96d?9a?bcd11112222c2?c1b2c3?c1c2c4?c1d2a按第二列分成二项2bcd第一项
abcd4a?44b?44c?44d?4abcd44446a?96b?96c?96d?99999?abcd222211114a4b4c4d4a?44b?44c?44d?46a6b6c6d?06a?96b?96c?96d?9
c3?4c2a2c4?6c2b2c3?4c2c2
第二项c4?9c2d210b?ab?ab?a2220c?a242424240d?add?a?ad222(4) 左边?aaa2424c?ac?ac?ac?a22224
b?ad?a?a222=
2b?a22
1d?ad(d?a)0d?b2b(b?a)c(c?a)1d(d?a)1c?a2=(b?a)(c?a)(d?a)2b?ab(b?a)
c(c?a)=(b?a)(c?a)(d?a)?
1b?ab(b?a)220c?bc(c?a)?b(b?a)22
2d(d?a)?b(b?a)=(b?a)(c?a)(d?a)(c?b)(d?b)?
1(c?bc?b)?a(c?b)221(d2?bd?b)?a(d?b)2
=(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
6
当n?2时,D2?xa2?1x?a1?x?a1x?a2,命题成立.
2假设对于(n?1)阶行列式命题成立，即 Dn?1?xn?1?a1xn?2???an?2x?an?1,
则Dn按第1列展开:
?1Dn?xDn?1?an(?1)n?10?1?1????00?x00??1x?1?xDn?1?an?右边
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 an1D1??a11??ann?， D2?a1nn(n?1)a1n?a11?ann? ，D3?an1ann?an1?a1n?， a11??证明D1?D2?(?1)2D,D3?D. 证明 ?D?det(aij) an1?D1??a11??ann?a1na11?(?1)n?1a11?(?1)n?1??a1nann?a2nan1?a21
???a1na2nann?? ?a3n?a1n? ?2(?1)n?2a21an1?a31?a11?(?1)n?1(?1)n?2?(?1)?an1annn(n?1)?(?1)1?2???(n?2)?(n?1)D?(?1)D
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n(n?1)a11?a1n?an1?n(n?1)n(n?1)同理可证D2?(?1)2?(?1)2DT?(?1)2D
?annn(n?1)n(n?1)2n(n?1) D3?(?1)
2D2?(?1)(?1)2D?(?1)n(n?1)D?D
7.计算下列各行列式（Dk为k阶行列式）：
a1?1a(1)Dn?,其中对角线上元素都是a，未写出的元素都是0；
xax?aan????aa?xn(2)Dn?a?a;
(a?1)(a?1)??????(a?n)(a?n)?n(3) Dn?1a??a1n?1n?1n?1; a?11a?n1提示：利用范德蒙德行列式的结果．
an?0a1c1?cn0b1d1?dn?0; bn(4) D2n?0(5)Dn?det(aij),其中aij?i?j;
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1?a111?a2?1????11?1?an(6)Dn?1?1,其中a1a2?an?0.
解
a00a0?0000a?00??????000?a0100按最后一行展开?0a(1) Dn?0?01
0a(?1)n?100a?0000?0?????000?a100?0(n?1)?(n?1)a?(?1)2n0?0?a?a(n?1)(n?1)
(再按第一行展开)
a?(?1)n?1?(?1)n?a(n?2)(n?2)?a?a?annn?2?an?2(a?1)
2(2)将第一行乘(?1)分别加到其余各行，得
xa?xDn?a?x?a?xax?a0?0a0x?a?0????0a00?x?a
再将各列都加到第一列上，得
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x?(n?1)a0Dn?0?0ax?a0?0n?1a0x?a?0????0a00?x?a
?[x?(n?1)a](x?a)
(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 行经(n?1)次对换换到第2行…，经n?(n?1)???1?交换，得
1n(n?1)n(n?1)2次行
1a?1?(a?1)n?1n??????1a?nDn?1?(?1)2a?an?1n
n?1n(a?n)a(a?1)(a?n)此行列式为范德蒙德行列式
n(n?1)Dn?1?(?1)n(n?1)2?[(a?i?1)?(a?n?1?i?j?1n(n?1)j?1)]
?(?1)?2?[?(i?n?1?i?j?1n?(n?1)???1j)]?(?1)2?(?1)2??[(i?n?1?i?j?1j)]
?(i?n?1?i?j?1j)
an?a1c1?cn00?b1d1?dn10
bn(4) D2n?0