4.计算下列各行列式:
(1)⎥⎥⎥⎥
⎦⎥⎢⎢⎢
⎢⎣⎢71
10
025*********
4; (2)⎥⎥⎥⎥⎦⎥⎢⎢⎢⎢⎣⎢-26
52321121314
1
2; (3)⎥⎥⎦⎥⎢⎢⎣⎢---ef cf bf de cd bd ae ac ab ; (4)⎥⎥⎥⎥⎦
⎥⎢⎢⎢
⎢⎣⎢---d c b a
1
00
110011001
解
(1)
71100251020214214
34327c c c c --0
10014
2310202110
21
4---=3
4)1(1431022
11014+-⨯---=14
31022110
14-- 3
21132c c c c ++14
171720010
99-=0
(2)
2605
232112131
412-24c c -2605032122130
412-24r r -0412032122130
412- 14r r -0
000032122130412-=0
(3)ef cf bf de cd bd ae ac ab ---=e
c b e c b e c b adf ---=1111111
11---adfbce =abcdef 4
(4)
d c b a 100110011001---21ar r +d
c b a
ab 1001
100
110
10---+=12)1)(1(+--d
c a ab 1011
1--+
2
3dc c +0
10111-+-+cd c ad
a a
b =23)1)(1(+--cd
ad
ab +-+111=1++++ad cd ab abcd
5.证明: (1)1
11222
2b b a a b ab a +=3)(b a -; (2)bz ay by ax bx az by ax bx az bz ay bx az bz ay by ax +++++++++=y x z x z y z y x b a )(33+;
(3)0)3()2()1()3()2()1()3()2()1()3()2()1(2
2222222
2
2222222
=++++++++++++d d d d c c c c b b b b a a a a ; (4)444422221111d c b a d c b a d c b a ))()()()((d b c b d a c a b a -----=))((d c b a d c +++-⋅;
(5)1
22
110000
0100001a x a a a a x x x n n n +-----
n n n n a x a x a x ++++=--111 . 证明
(1)0
0122222221
312a b a b a a b a ab a c c c c ------=左边a b a b a b a ab 22)
1(22213-----=+21))((a b a a b a b +--= 右边=-=3)(b a
(2)bz ay by ax z by ax bx az y bx az bz ay x a ++++++分开
按第一列
左边
bz
ay by ax x by ax bx az z bx
az bz ay y b +++++++ ++++++002y by ax z x bx az y z bz ay x a 分别再分
bz
ay y x by ax x z bx az z y b +++z y x y x z x
z y b y x z x z y z y x a 33+分别再分
右边=-+=233)1(y
x z x z y z
y x b y x z x z y z y x a
(3) 22
2
22222
2222
2
222
)3()2()12()3()2()12()3()2()12()3()2()12(++++++++++++++++=d d d d d c c c c c b b b b b a a a a a 左边9
644129644129
644129644122
2221
41312++++++++++++---d d d d c c c c b b b b a a a a c c c c c c 9644964496449644222
2
2
++++++++d d d d c c c c
b b b b a a a a 分成二项按第二列9
64
41964419
644196441222
2+++++++++d d d c c c b b b a a a 94
94949494642
2
22
24232423d d c c b b a a c c c c c c c c ----第二项
第一项
06416416416412
22
2=+d
d
d c c c b
b b a a a (4) 4
44444422222220
001a
d a c a b a a
d a c a b a a
d a c a b a ---------=左边=)()()(222222222222222a d d a c c a b b a d a c a b a
d a c a b --------- =)
()()(1
11))()((222a d d a c c a b b a d a c a
b a d a
c a b ++++++--- =⨯---))()((a
d a c a b )
()()()()(0
0122222a b b a d d a b b a c c a b b b
d b c a b +-++-++--+ =⨯
-----))()()()((b d b c a d a c a b )
()()()(1
122
22
b d a b bd d b
c a b bc c ++++++++ =))()()()((
d b c b d a c a b a -----))((d c b a d c +++-
(5) 用数学归纳法证明
.,1
,22121
22命题成立时当a x a x a x a x D n ++=+-=
=
假设对于)1(-n 阶行列式命题成立,即 ,122111-----++++=n n n n n a x a x a x D
:1列展开按第则n D
1
1
1
010001
)1(1
1----+=+-x
x
a xD D n n n n
右边=+=-n n a xD 1 所以,对于n 阶行列式命题成立.
6.设n 阶行列式)det(ij a D =,把D 上下翻转、或逆时针旋转 90、或依副对角线翻转,依次得 n nn n a a a a D 11111 =, 11112n nn n a a a a D = ,11
113a a a a D n n
nn =,
证明D D D D D n n =-==-32
)1(21,)1(.
证明 )det(ij a D =
n
nn
n n
n n nn n a a a a a a a a a a D 22111111111
1
1)1(
--==∴
=--=--n
nn n n
n
n n a a a a a a a a 3311
22111121)1()1( nn
n n n n a a a a 111121)1()1()1(---=--D D n n n n 2)
1()
eoa1()2(21)1()1(--+-+++-=-=
同理可证nn n n n n a a a a D 11112
)1(2)
1(--=D D n n T
n n 2)
1(2)1()1()1(---=-= D D D D D n n n n n n n n =-=--=-=----)1(2
)1(2
)1(22
)1(3)1()
1()
1()1(
7.计算下列各行列式(阶行列式为k D k ):
(1)a a
D n 1
1
=
(2)x
a
a
a
x a
a a x D n
=
; (3) 1
1
11)()1()()1(11
11
n a a a n a a a n a a a D n n n n n
n n ------=---+; 提示:利用范德蒙德行列式的结果. (4) n
n
n
n
n d c d c b a b a D
000
011112=; (5)j i a a D ij ij n -==其中),det(;
(6)n
n a a a D +++=
11
11
111
112
1 ,021≠n a a a 其中.
解
(1) a
a a a a D n 000100000000
00001000 =