高等代数习题库
第一章 行列式
1. 决定以下排列的反序数,从而决定它们的奇偶性. (1) 134782695 (2) 217986354 (3) 987654321
2. 如果排列x1x2?xn?1xn的反序数为k,排列xnxn?1?x2x1的反序数是多少? 3. 写出4阶行列式中所有带有负号并且包含因子a2a3的项. 4. 按定义计算行列式
0010?0002?00????00?n?1000??02???0010?0000? 0n(1) ?0n; (2)
?n?10x1x2111x2xa1j2a2j2?anj22?111??a1jna2jn??anjn5. 设 f(x)?131a1j1,不计算行列式,求展开式中x3的系数.
6. 求
?j1j2?jna2j1?anj1,这里
?j1j2?jn是对所有n元排列求和.
7. 证明:
a11(t)a11(t)da21(t)dt?an1(t)a12(t)a22(t)?an2(t)???a1n(t)a2n(t)?ann(t)n??ddtddt?ddta1j(t)a2j(t)??a1n(t)a2n(t)???j?1a21(t)?an1(t)
?anj(t)?ann(t)8. 计算下列行列式.
246427543721327443; (2) 621xyx?yyx?yxx?yxy123412222341241232222(1) 1014?342; (3)
234
1?x11?x11111?y111101111?y73?9?2?324736934a2222(a?1)(b?1)(c?1)(d?1)2222(a?2)(b?2)(c?2)(d?2)12235?74(a?3)(b?3)(c?3)(d?3)410(4)
11101011; (5)
bcd
11013751321 210(6)
111; (7)
875; (8) 321111675739. 已知n阶行列式
a11D?a21?an1b1,b2,?,bn为常数,若Da12a22?an2????a1na2n?ann
的值为c,求下列行列式的值:
a11b1?an1bnb12a12b1b2a22b2?an2bnb22????a1nb1bna2nb2bn?annbn2a21b2b1
10. 设D是n阶行列式,若D的元素间满足关系:
aij??aji(i,j?1,2,?,n)
则称D是一个反对称行列式. 求证:当n是奇数时, n阶反对称行列式的值为零.
11. 计算下列n阶行列式
1112103???1012222223???222; ?22?n(1) 10?100; (2) 2??20?nx?aaax?aaaax?a???aaa?(3)
a?a;
aa?x?a1?a1111?a21111?a31???111?(ai?0,i?1,2,?,n);
(4)
1?11?1?an2a1a2aa2n(a?1)(a?1)?a?11n??(a?n)(a?n)?n2a1a??1a2n?1n?1n?1(5)
?? ; (6)
2a?a1;
??a?n1x0yx0y??0000?; (8)
1?a1a1a1?a1a21?a2a2a2a3a31?a3a3???ananan?1?an(7) ?0y0000??x0
yx?12. 证明
b?cc?ac1?a1c2?a2yz0xzyx0?01110a?babb1b2cc1; c2(1) b1?c1b2?c20x0zya1?b1?2a1a2?b210zy22a21z21y