大一线性代数练习题五套(带答案)

01A Am n,Bn m(m n), n ( ).

(A)BA;(B)AB;(C)(BA)T;(D)ATBT.

01B , ( ).

(A)(AB)T ATBT;

(B) A B, |A| |B|;

(C) A,B , A B ;

(D)A2 E2 (A E)(A E).

01C , ( ).

(A) A n , (A E)(A E) (A E)(A E);

(B) A,B n 1 , ATB BTA;

(C) A,B n , AB 0, (A B)2 A2 B2;

(D) A n , AmAk AkAm.

01D n (1/2,0, ,0,1/2), A E T ,B E 2 T , E n , AB ( ).

(A)O;(B) E;(C)E;(D)E T .

O ( ). 1 B AT01E A,B n , ( 3) O

(A)( 3)n|A||B 1|;

(C) 3|AT||B 1|;(B) 3|AT||BT|;(D)( 3)2n|A||B 1|.

01F A B n , A(B E) O, ( ).

(A)A O B E;

(C)detA 0 detB 1;(B)detA 0 det(B E) 0;(D)A BA.

01G A,B n , ( ).

(A)|A B| |A| |B|;

(B)|A B| |A| |B|;(C) AB B, AB BA;(D) AB B E, AB BA.

.9.

01H A,B n , , ( ).

(A)AB O A O B O;

(B)|A| 0 A O;

(C)|AB| 0 |A| 0 |B| 0;

(D)A E |A| 1.

01I 1 A n , A 3, A _________.

3 A 3 2 ,B 2 , , , 2, 3 3 , 2 3 3 01J

|A| 24,|B| 3, |A B|.

01K A n , AAT E,|A| 0. |A E|.

01L (1,0, 1)T, A T. |aE An|, E 3 ,n .

01M A (aij) n ,Aij aij , A ,

A22A23 A2n

A32A33 A3n a11|A|n 2 |D|

An2An3 Ann

01N A n , A , (A B)2 E,

(E A 1BT)T(E BA 1) 1.

01O A n ,B n , A B2 .

01P A,B n , A B E. :AB BA.

01Q A,B n ,A2 A,B2 B,(A B)2 A B,

AB O.

01R A2 E, A . A,B , : AB A B ( AB BA).

.10.

2 102E A 0 0400 200 n , A.024 002

100

5 2 10 ,AA. 211 100 ,P 02F AP PB, B 000 00 1

cos 02G sin sin cosn sinn cos n sinn .cosn

02H A E T, E n , n , T , :

(1)A2 A T 1;

(2) T 1 ,A .

03A

(1) A 1,(A) _________. 000 n 1 n00 0

0 0 ,B (E A) 1(E A), (E B) 1 ________.0 7

1010 002 00 00 1 230 (2) A 0 45 0 6 0(3) A n , (A E)2 2(A E)2, A ___________.

(4) A,B ,E ,

0 2 1 AB A 2B,B 0 10 , 201

(A 2E) 1 __________.

(5) n A,B A*BA 4BA 2E, |A| 2,|E 2A| 0, B ___________.

.13.

03B A,B n , AB A B A E .

112 A 035 , B. 101

03C n A A3 A2 A E 0, A , A 1.03D A A2 2A 3E 0, (A 4E) 1.

03E A A2 3A 10E O, :A,A 4E , .

03F A3 E,B E 2A A2, B , .

03G A,B n ,B E AB . :E BA ,

1 (E BA).

03H A,B n ,A3 2E,B A2 2A E, E n

1 . :B , B.

03I A,B n ,E BA , E n ,

1 :E AB , (E AB).

03J n A (aij) aij 0(i,j 1,2, ,n) aij aii

j 1j in

(i 1,2, ,n), A .

03K A2 B2 E, |A| |B| 0, :A B .

03L A m n ,B n m , m n, AB .03M A , :A 1 |A| 1.

1 103N A 2 1 000 200 1 A. 130 214

2 12 03O A 21 2 . 2 21

.14.

03P A 00021 00053 12300 , A 1.45800 34600

A2 .A4 A103Q A1,A4 m,n , A A3

04A A , |A| 3,

(1)|2A 1| _________;

(3)|(A*)*| ________;

(5)|5A 1 2A*| _________;

(7)|4A (A*)*| _________.(2)|A*| _________;(4)|(A*) 1| __________;(6)|(2A)*| ________;

11 A (3A)*=04B A 3 , |A| 2, 12 ||.

04Cn , A2 E, B (Aij)n n, Aij |A| aij , B2 E.

04D .

04E A n , :(A ) |A|n 2A.

04F 3 A A

.

04G A n , n ,b ,

0 EP ,T*A A AQ T 1 111 121 A* 113 ,b

A* A ,E n .

(1) PQ;

(2) : Q TA 1 b.

.15.

100 a11a12a13 a11a13a12 05A A a21a22a23 ,B 2a212a232a22 ,P1 001 , a a 010aaaa3332 33 3132 31

100 P2 020 , B ( ). 001

(A)P1P2A;(B)AP2P1;(C)P1AP2;(D)P2AP1.05B n A n , A .05C

20 1 062 A 1113 1 19 71 410 .616 14 34 0

05D A , a,b ,

A

05E

1 0A 2 3 1111 1 121 3a 24b 3 515 c 8 11 01 1b .a4 23 3517 11

a,b,c . A .

05F n

a b

A b b

n 2. A .

.16.bab bbba b b b , b a

1 a

05G n( 3) A a ... a a a a n 1, a. ......... aa...1 aa...1a...a1...

05H A 4 ,A* A , r[(A*)*].

05I A n ,B n m ,r(B) n AB O, :A O.05J

A BC.

05K

0111 1011 A ,1 10 1 1 1 10

** AXA 3XA 12E, E 4 ,A A , A n , B C( C2 C), X.

05L A

1/201 01/20*A 001/2

0000 2 , 0 1

1 1 X 3AXA XA 2E, E 4 . X.

05M

1 1A 0 0 0 0 ,2 0 10 230000

1* 1 1 B [(A)]BA 2AB 12E, E 4 . B.2

.17.

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