大一线性代数练习题五套(带答案)

(D) E D TD.
01E
䆒 A , B 䛑ᰃ n 䰊ৃ䗚䰉, ߭
(3)
¨§ ¨©
AT O
O B
1¸¸¹·
(
).
(A) (3) n | A || B 1 |;
(B) 3 | AT || B T |;
(C) 3 | AT || B 1 |;
(D) (3)2n | A || B1 | .
(5) 䆒 n 䰊ᮍ䰉 A , B ⒵䎇 A *BA 4BA 2E , Ϩ | A | 2 , | E 2A| z 0 ,
߭ B ___________.
. 13 .
03B 䆒 A , B ഛЎ n 䰊ⶽ䰉, ⒵䎇 AB A B 䆕ᯢ A E ৃ䗚.
§1 1 2·
བᵰ A
¨ ¨¨
0
3
5
04F Ꮖⶹ 3 䰊ⶽ䰉 A ⱘ䗚ⶽ䰉Ў A 1
§1
¨ ¨ ©
1 1
1 2 1
1·
1 3
¸ ¸ ¹
䆩∖Ԉ䱣ⶽ䰉
A * ⱘ䗚
ⶽ䰉.
04G 䆒 A Ў n 䰊䴲༛ᓖⶽ䰉 , D Ў n 㓈߫৥䞣 , b Ўᐌ᭄ , 䆄ߚഫⶽ䰉
§ E 0·
§A D·
P
¨ ©

D
T
A
*
A
¸, ¹
Q
¨©D T
b
¸ ¹
,
݊Ё A *ᰃⶽ䰉 A ⱘԈ䱣ⶽ䰉 , E Ў n 䰊ऩԡⶽ䰉 .
0 # 0
2 # 0
" "
©n 0 0 "
0·
0¸
¸
n
#
1¸¸¸
,
߭
( A )1
¸
0¹
_________ .
§ 1 0 0 0·
(2) 䆒 A ¨¨ 2 3
0
0
¸ ¸
,
B
(E A)1(E A), ߭ (E B)1
________.
¨¨ 0 4 5 0 ¸¸
© 0 0 6 7¹
䰉. 䆕ᯢ : B ᰃৃ䗚ⶽ䰉, ᑊ∖ B 1.
03I 䆒 A, B ᰃ n 䰊ⶽ䰉, E BA ᰃৃ䗚ⶽ䰉 , ݊Ё E ᰃ n 䰊ऩԡⶽ䰉,
䆕ᯢ: E AB ৃ䗚 , ᑊ∖(E AB )1.
n
03J Ꮖⶹ n 䰊ᅲⶽ䰉 A ( a ij ) Ё a ij t 0 (i, j 1 , 2 , " , n ) Ϩ ¦ aij aii
∖ᅗӀⱘ䗚ⶽ䰉 . 03F Ꮖⶹ A3 E, B E 2 A A 2 , 䆕ᯢ B ৃ䗚 , ᑊ∖ߎ݊䗚䰉 . 03G 䆒 A, B Ў n 䰊ⶽ䰉, B Ϣ E AB ഛৃ䗚. 䆕ᯢ: E BA гৃ䗚 ,
ᑊ∖(E BA)1. 03H 䆒 A, B Ў n 䰊ⶽ䰉 , A 3 2E, B A 2 2 A E , ݊Ё E Ў n 䰊ऩԡⶽ
05C ∖ϟ߫ⶽ䰉ⱘ髟
§1 2 0 0 1·
A
¨ ¨
0
¨ ¨
1
6 11
2 3
4 6
10 16
¸ ¸ ¸ ¸
.
© 1 19 7 14 34 ¹
05D ∖ⶽ䰉 A ⱘ髟 , ݊Ё a , b ᰃখ᭄ ,
§1 1 1 1·
A
¨ ¨
0
1
1
b
¸ ¸
.
¨2 3 a 4¸
¨
¸
©3 5 1 7¹
05E 䆒ⶽ䰉
(1) 䅵ㅫᑊ࣪ㅔ PQ ;
(2) 䆕ᯢ : ⶽ䰉 Q ৃ䗚ⱘ‫ܙ‬㽕ᴵӊᰃ D T A 1 D z b.
. 15 .
05A 䆒 A
§a ¨¨a
11 21
©¨a 31
a 12 a 22 a 32
a13 a 23
· ¸ ¸
,
B
a
33
¸ ¹
§ ¨ ¨
a11 2a 21
¨ ©
a 31
a 13 2a 23 a 33
㓐ড়㒗дѠ
01A 䆒 Amu n , Bn um (m z n), ߭ϟ߫䖤ㅫ㒧ᵰϡЎ n 䰊ᮍ䰉ⱘᰃ ( ).
(A) BA ;
(B) AB ;
(C) (BA)T ;
(D) AT B T .
01B ೼ϟ߫ੑ乬Ё, ℷ⹂ⱘᰃ( ).
(A) ( AB )T AT BT ;
(B) 㢹 A z B, ߭ | A | z | B |;
2aa1222¸·¸ , P1
a 32
¸ ¹
§1
¨
¨¨ ©
0 0
0 0 1
0·
¸
1 0
¸¸ ¹
,
§1 0 0·
¨
¸
P2 ¨ 0 2 0 ¸ , ߭ B ( ).
¨ ©
0
0
1
¸ ¹
(A) P1 P2 A;
(B) AP2 P1 ;
(C) P1AP2 ;
(D) P2 AP1 .
05B 䆕ᯢབᵰ n 䰊ⶽ䰉 A Ϣӏᛣ n 䰊ⶽ䰉䛑ৃѸᤶ , 䆕ᯢ A ᰃ᭄䞣ⶽ䰉.
䰉 , n Ўℷᭈ᭄.
01M 䆒 A (aij ) ᰃ n 䰊ⶽ䰉, Aij ᰃ aij ⱘҷ᭄ԭᄤᓣ , 㢹 A ᰃৃ䗚䰉, 䆩
䆕ᯢ | D |
A22 A 23 " A 2n A32 A 33 " A3n
" " "" An2 A n3 " Ann
a11 | A | n 2
01N Ꮖⶹ A ᰃ n 䰊ᇍ鹵ⶽ䰉, Ϩ A ৃ䗚, བ ( A B) 2 E, ࣪ㅔ
2 , ߭
§1 ©12
A· ¹
1

(
3
A
)*
=
.
04C n 䰊ⶽ䰉, ⒵䎇 A2 E , জⶽ䰉 B ( A ij ) nun , ݊Ё A ij ᰃ㸠߫ᓣ | A |
Ё‫ܗ‬㋴ aij ⱘҷ᭄ԭᄤᓣ , 䆕ᯢ B 2 E .
04D 䆩䆕Ϟϝ㾦ⶽ䰉ⱘԈ䱣ⶽ䰉ҡЎϞϝ㾦ⶽ䰉 .
04E 䆒 A Ў n 䰊ৃ䗚ⶽ䰉 , 䆕ᯢ : ( A ) | A | n2 A .
(A) | A B | | A| | B |;
(C) 㢹 AB B, ߭ AB BA ;
(B) | A B | | A | | B |;
(D) 㢹 AB B E, ߭ AB BA. .9.
01H 䆒 A , B ᰃ n 䰊ⶽ䰉 , ߭೼ϟ߫ੑ乬Ё, ℷ⹂ⱘᰃ( ). (A) AB O ⱘ‫ߚܙ‬ᖙ㽕ᴵӊᰃ A OϨ B O ; (B) | A| 0 ⱘ‫ߚܙ‬ᖙ㽕ᴵӊᰃ A O ; (C) | AB | 0 ⱘ‫ߚܙ‬ᖙ㽕ᴵӊᰃ | A| 0 ៪ | B | 0 ; (D) A E ⱘ‫ߚܙ‬ᖙ㽕ᴵӊᰃ | A | 1. 01I 䆒 A ᰃ n 䰊ৃ䗚ⶽ䰉, བᵰ A Ё↣㸠‫ܗ‬㋴П੠䛑ᰃ 3, 䙷М A1 ↣㸠 ‫ܗ‬㋴П੠ᖙᰃ _________ .
(D) བ A ᰃ n 䰊ⶽ䰉, ߭ Am A k Ak A m .
01D 䆒 n 㓈㸠৥䞣 D (1/ 2 , 0 ,", 0 , 1/ 2 ), ⶽ䰉 A E D TD , B E 2D TD,
݊Ё E Ў n 䰊ऩԡⶽ䰉 , ߭ AB ㄝѢ ( ).
(A) O ;
(B) E ;
(C) E ;
(C) 䆒 A , B ᰃϝ㾦ⶽ䰉 , ߭ A B гᰃϝ㾦ⶽ䰉 ;
(D) A2 E 2 ( A E )( A E ).
01C ೼ϟ߫ੑ乬Ё, ϡℷ⹂ⱘᰃ ( ).
(A) བ A ᰃ n 䰊ⶽ䰉 , ߭ ( A E )(A E ) ( A E )( A E );
(B) བ A , B ഛᰃ n u1 ⶽ䰉 , ߭ AT B B T A ; (C) བ A , B ഛᰃ n 䰊ⶽ䰉, Ϩ AB 0, ߭ ( A B) 2 A2 B 2 ;
01F 䆒 AǃB ഛЎ n 䰊ᮍ䰉, Ϩ A(B E ) O, ߭ ( ).
(A) A O ៪ B E ;
(B) det A 0 ៪ det( B E ) 0 ;
(C) det A 0 ៪ det B 1;
(D) A BA.
01G 䆒 A, B Ў n 䰊ⶽ䰉, ϟ߫㒧䆎ℷ⹂ⱘᰃ ( ).
¨§ D ¸·
¨§ E ¸·
01J
䆒᳝ 3 䰊ⶽ䰉 A
¨3J 2 ¸ , B
¨ ©
2J
3
¸ ¹
¨ ¨¨©
J J
2 3
¸ ¸¸¹
,
݊Ё
D
,
E
,
J
2
,J
3
ഛЎ 3
㓈㸠৥䞣,
ϨᏆⶹ㸠߫ᓣ | A| 24, | B | 3, ∖ | A B | .
01K 䆒 A Ў n 䰊ⶽ䰉, Ϩ⒵䎇 AAT E, | A | 0. ∖ | A E | . 01L 䆒 D (1,0, 1)T , ⶽ䰉 A DD T. 䅵ㅫ | aE A n |, ݊Ё E Ў 3 䰊ऩԡⶽ
( E A1BT )T (E BA 1)1.
01O Ꮖⶹ A ᰃ n 䰊ᇍ鹵ⶽ䰉, B ᰃ n 䰊ডᇍ鹵ⶽ䰉, 䆕ᯢ A B 2ᰃᇍ鹵
ⶽ䰉 .
01P 䆒 A , B Ў n 䰊ⶽ䰉, Ϩ A B E. 䆕ᯢ: AB BA .
01Q Ꮖⶹ A , B ഛᰃ n 䰊ⶽ䰉 , A2 A, B 2 B, ( A B)2 A B,
02H 䆒 A E [[ T , ݊Ё E ᰃ n 䰊ऩԡⶽ䰉, [ ᰃ n 㓈䴲䳊߫৥䞣 , [ T ᰃ
[ ⱘ䕀㕂, 䆕ᯢ :
(1) A2 A ⱘ‫ܙ‬㽕ᴵӊᰃ [[ T 1 ;
(2) ᔧ [ T[ 1 ᯊ, A ᰃϡৃ䗚ⶽ䰉 .
03A ฿ぎ乬
§0 1 0 "
(1) Ꮖⶹ A
¨0
¨
¨¨¨ ¨
# 0
¨
¸
©0 0 0 2 ¹
02F Ꮖⶹ AP
PB, ݊Ё B
§1 0 0 ·
¨ ¨¨©
0 0来自百度文库
0 0
0 1
¸ ¸¸¹
,
P
§1 0 0 ·
¨ ¨
2
1
0
¸ ¸
,
∖
A
ঞ A5.
¨© 2 1 1 ¸¹
§ cosT sinT ·n § cos nT sin nT · 02G 䆕ᯢ ¨¨©sinT cosT ¸¸¹ ¨¨©sin nT cos nT ¸¸¹ .
(3) | (A* )*| ________;
(4) | ( A*)1 | __________;
(5) | 5A 1 2A *| _________;
(6) | (2A )* | ________;
(7) | 4A ( A* )* | _________.
| | 04B 䆒 A Ў 3 䰊ⶽ䰉 , Ϩ | A |
¨§ 1 1 1
1
1 ¸·
A
¨ 0 1 1 2 1 ¸
¨ ¨¨©
2 3
3 5
a2 1
4 c8
b
5
3
¸ ¸¸¹
݊Ё a , b, c Ўখ᭄ . ∖ⶽ䰉 A ⱘ髟 .
05F 䆒 n 䰊ⶽ䰉
¨§ a b b " b ¸·
¨b a b " b¸
A
¨ ¨
b
b
a
"
b
¸ ¸
,
¸ ¸¸
,
∖ⶽ䰉 B .
©1 0 1¹
03C 䆒 n 䰊ⶽ䰉 A ⒵䎇݇㋏ᓣ A3 A2 A E 0, 䆕ᯢ A ৃ䗚 , ᑊ∖ A1.
03D Ꮖⶹⶽ䰉 A ⒵䎇݇㋏ᓣ A 2 2 A 3E 0 , ∖ ( A 4 E ) 1.
03E 䆒ᮍ䰉 A ⒵䎇ᮍ⿟ A2 3A 10E O, 䆕ᯢ: A , A 4E 䛑ৃ䗚 , ᑊ
¨§ 1 0 0 0 ¸·
03N ∖ⶽ䰉 A
¨1
¨ ¨¨©
2 1
2 1 2
0 3 1
0¸
0 4
¸ ¸¸¹
ⱘ䗚䰉 A1 .
¨§ 1 2 2 ¸· 03O ∖ A ¨ 2 1 2 ¸ ⱘ䗚ⶽ䰉 .
¨© 2 2 1 ¸¹
. 14 .
§0 0 0 2 1·
¨¨ 0 0 0 5 3 ¸¸
¨
¸
03P 䆒 A ¨ 1 2 3 0 0 ¸ , ∖ A 1 .
䆕ᯢ AB O. 01R བᵰ A 2 E, ߭ A 鹵Ўᇍড়ⶽ䰉. 䆒 A, B 䛑ᰃᇍড়ⶽ䰉 , 䆕ᯢ: Ь鳥
AB ᰃᇍড়ⶽ䰉ⱘ‫ߚܙ‬Ϣᖙ㽕ᴵӊᰃ A Ϣ B ৃѸᤶ ( ेAB BA ). . 10 .
§2 4 0 0 ·
02E Ꮖⶹ A
¨ ¨¨¨
1 0
2 0
0 2
0 4
¸ ¸¸¸
,
∖
A n.
j1 jzi
(i 1, 2 ," , n), 䆩䆕 A ᰃৃ䗚ⶽ䰉 .
03K 㢹 A2 B 2 E , Ϩ | A| | B | 0 , 䆩䆕ᯢ : A B ᰃϡৃ䗚ⶽ䰉.
03L 䆒 A Ў m u n ⶽ䰉 , B ᰃ n u m ⶽ䰉, Ϩ m ! n , 䆕ᯢ AB ᰃϡৃ䗚ⶽ䰉. 03M 㢹ⶽ䰉 A ⱘ‫ܗ‬㋴ഛЎᭈ᭄ , ∖䆕 : A 1 ⱘ‫ܗ‬㋴ഛЎᭈ᭄ⱘ‫ܙ‬㽕ᴵӊЎ | A| r1.
(3) Ꮖⶹ A ᰃ n 䰊ⶽ䰉, Ϩ( A E )2 2( A E) 2, ߭ A 1 ___________ .
(4) 䆒 A, B ഛЎϝ䰊ⶽ䰉, E ᰃϝ䰊ऩԡⶽ䰉 , Ꮖⶹ
§ 1 0 2·
AB
A 2B, B
¨ ¨¨
0
1
0
¸ ¸¸
,
© 2 0 1 ¹
߭ ( A 2E) 1 __________ .
¨ ¨¨
4
5
8
0
0
¸ ¸¸
©3 4 6 0 0¹
03Q 䆒 A1 , A 4 ߚ߿Ў m , n 䰊ৃ䗚ⶽ䰉 , 䆩∖ A
§ ¨
A1
¨ ©
A3
A
2
· ¸
ⱘ䗚ⶽ䰉.
A
4
¸ ¹
04A 䆒ⶽ䰉 A Ўϝ䰊ৃ䗚ⶽ䰉 , Ϩ| A | 3 , ߭
(1) | 2A 1| _________;
(2) | A *| _________;