线性代数 第二版 陈殿友 课后答案.khdaw.pdf

1.1(A) 2.a1,a2b1,b2,b31, 3.b1,b2,b3c1,c2,2, 4. . aijaibj, 301 213 ; bijbicj, 21 43 12 . 1.2(A) 1. : ; 204 111 220 131 4311 0412 )2 ; 11 )5 b a b a mbma ba , 2 2 10520 876 204 131 210 131 4311 0412 )2 00 00 11 )5 b a b a mbma ba 2. , 111 111 111 A , 150 421 321 B 3AB- 2AATB. 111 111 111 AB 150 421 321 092 650 850 222 222 222 0276 18150 24150 23AAB 2294 20172 22132 111 111 111 TB A 150 421 321 092 650 850 3.A=PQ, 2, 1, 2, 1 2 1 QP AA100. 1 12 2 50 0 0 2 1818 2424 0 0 6 2 2 2222 22 2 2929 20172017 2222 212 424 212 2, 1, 2 1 2 1 PQA 2 1 2 1 2, 1, 2Qp QPQpQQPPPQA)2()2()()( 999999100100 APQ 9999 2)(2 . 212 424 212 299 6.4(A) 2. 3221 2 3 2 2 2 1321 2432),()1xxxxxxxxxxf 3221 2 3 2 2 2 1321 2435),()2xxxxxxxxxxf 3221 2 3 2 2 2 1321 44543),()3xxxxxxxxxxf : 1) , 310 122 021 A , 02 22 21 2 A . 2) 6.4(A)6.4(A)6.4(A)6.4(A) 2 2 2 x2 2323 2 2 2 2 2 1 1 353 2 1 2 2 3)3 2 2 1 100 020 003 100 022 025 110 132 025 12 12 23 23 rr cc rr cc A . 3) , 520 242 023 A , 08 42 23 , 03 21 AA , 024 520 242 023 A . 3. 3221 2 3 2 2 2 1321 482),(xxxaxxxxxxxf , a, . : 424 2 2 0 0 , 0 , 024024 5 5 . . 820 222/ 02/1 a a A , 0 4 2 22/ 2/1 , 01 2 21 a a a AA ,22a , 0212 820 222/ 02/1 2 aa a A ,6a 6a ,. 1.3(A) 3. 2200 0200 0034 0043 A A4. 2200 0200 0034 0043 2200 0200 0034 0043 2 A . 1.3(A)1.3(A) 3 3 A A 4800 0400 00250 00025 4800 0400 00250 00025 4800 0400 00250 00025 4 A 816400 01600 006250 000625 1.4(A) 3.AB (1)A2 (2)AB-BA (3)ABAB=BA. (1) ,)()()( 222T2 AAAA T A2. (2) TTT )()()(BAABBAAB TTTT BAAB BAAB)()( BAAB AB-BA. (3) BAABABAB)()( TTT AB ABAB T )( AB=BA 1 1.4(A)1.4(A) AB )( AB=BA ABAB T )( AB ABAB=BA. 1.4(A) 3.AB (1)A2 (2)AB-BA (3)ABAB=BA. (1) ,)()()( 222T2 AAAA T A2. (2) TTT )()()(BAABBAAB TTTT BAAB BAAB)()( BAAB AB-BA. (3) BAABABAB)()( TTT AB ABAB T )( AB=BA AB=BA ABAB T )( AB ABAB=BA. 1.5(A) 1. 1403 1302 1201 )1 T T B B B B)()( A A BABA B)( A A ABAB T T )()ABAB 3403 1302 1201 31 21 3 2 rr rr 6200 3100 1201 )1( 2 r 6200 3100 1201 0000 3100 5001 32 12 2 2 rr rr 12433 02322 12211 34311 )3 810500 66300 22100 34311 12433 02322 12211 34311 41 31 21 3 2 rr rr rr 810500 66300 22100 34311 )1( 2r 20000 00000 22100 32011 42 32 12 5 3 3 rr rr rr 00000 10000 22100 32011 34 4 2 1 rr r 00000 10000 02100 02011 23 13 2 3 rr rr 2.1(A) 1. 0000 311 1 5050 63063 212 3434 0 0 0 10010 4313 381 141 102 )1( )1()1(03)4(2 381 141 102 8)1(2310)1()4(1811 =- 4 3.ij817i25j49 (817325649)=7+0+5+1+0+1+1+0=15 817325649i=3, j=6 5. )1( 5541322413 aaaaa. )2( 5524134231 aaaaa (1) (34215)=2+2+1+0=5 5541322413 aaaaa (2) 55423124135524134231 aaaaaaaaaa (34125)=2+2+0+0=4 5524134231 aaaaa 2.2(A) 1. ; 4321 6510 0531 1021 )1(; 1111 1111 1111 1111 )2( ; )3( efcfbf decdbd aeacab 49)=7+0+5+1+0+1+1+0=15 49)=7+0+5+1+0+1+1+0=15 41 1a a aa41 415 )2( )2( (34215)=2+2+1+0=5 4231421 aaaaa 14231421 (34125)=2+2+0+0=4(34125)=2+2+0+0=4 4321 6510 0531 1021 )1( 3300 6510 1510 1021 41 21 r- r r- r 3300 7000 1510 1021 32 r- r 27 7000 3300 1510 1021 43 rr 1111 1111 1111 1111 )2(8 2000 0200 0020 1111 41 31 21 rr rr rr efcfbf decdbd aeacab )3( ecb ecb ecb adf abcdefabcdef4 111 111 111 2. yxz xzy zyx ba bzaybyaxbxaz byaxbxazbzay bxazbzaybyax )( )2( 33 : bzaybyaxaz byaxbxazay bxazbzayax bzaybyaxbx byaxbxazbz bxazbzayby bzaybyaxz byaxbxazy bxazbzayx a bzaybyaxx byaxbxazz bxazbzayy b aybyaxz axbxazy azbzayx a bzaybyx byaxbxz bxazbzy b e e e e abcdefabcdef4 41 1abcdefabcdef axbxabx bazbzaz ybyaxz xbxazy zbzayx a2 bzayyx byaxxz bxazzy b2 yaxz xazy zayx a2 bzyx byxz bxzy b2 yxz xzy zyx a3 zyx yxz xzy b3 yxz xzy zyx a3 yxz xzy zyx b3 yxz xzy zyx ba)( 33 2.2(B) 1. ; 12/12/12/1 2/112/12/1 2/12/112/1 2/12/12/11 )1( ; )3( abcd badc cdab dcba 12/12/12/1 2/112/12/1 2/12/112/1 2/12/12/11 )1( 2111 1211 1121 1112 16 1 2115 1215 1125 1115 16 1 4,3,2 1 j ccj 16 5 1000 0100 0010 1115 16 1 4, 3, 2 1 i rr i 2/2 , )3( abcd badc cdab dcba A abcd badc cdab dcba abcd badc cdab dcba AAT 2222 2222 2222 2222 000 000 000 000 dcba dcba dcba dcba 42222 2 )( A dcbaAAAA TT .)(A 22222 dcba 2.3(A) 1.4 . 100 110 011 001 )1( d c b a d c b a 100 110 011 001 d c b a 10 11 01 d c 10 10 011 )1(1 21 1)-(- cd-)( dbbcda 2 2 cb b2 2 .) .)2 22 ) )d 222 dcd 2 424 )d 2.3(A) 1 1 0101 00 c 1cdadababcd 2. . 0 , 111 111 111 )3( 21 2 1 n n n aaa a a a D () 1110 1110 1110 1111 2 1 n n a a a D 001 001 001 1111 2 1 . 1, 3, 2 1 n ni rr a a a i 000 000 000 111 1 1 2 1 1 ., 2, 1 1 11 n n ii ni cc a a a a a i i ) 1 1( 1 21 n ii n a aaa 2.3(B) 1. . 4 , 3 , 2 , 1 , 0, 0 , 0001 0001 0001 0001 1111 )2( 4 3 2 1 0 ka a a a a a B k n n a 11 0 0 0 0 a 1 1(1 n n 0000 0000 0000 0000 1111 1 4 3 2 1 4 1 0 . 43, 2, 1 1 11 a a a a a a B ii i cc a i i ) 1 ( 4 1 04321 ii a aaaaa 2.Laplace 0 0 11110 00321 00111 )1( 2 4 2 3 2 2 2 1 4321 xxxx xxxx 1,2 2 4 2 3 2 2 432 2121 111 )1( 21 11 xxx xxx 2 4 2 3 2 1 431 3121 111 )1( 31 11 xxx xxx )()( 232434 xxxxxx )()(2 131434 xxxxxx : 5.1,4,. 3.1(A) 3.nAA3=0E- A(E- A)- 1=E+A+A2 A3=0 2 2 2 2 111 x x xx2 (E- A)(E+A+A2)=E3- A3=E. (E+A+A2)(E- A)=E3- A3=E. E- A(E- A)- 1=E+A+A2 3.1(A) 3.nAA3=0E- A(E- A)- 1=E+A+A2 A3=0 (E- A)(E+A+A2)=E3- A3=E. (E+A+A2)(E- A)=E3- A3=E. E- A(E- A)- 1=E+A+A2 3.2(A) 2(2). 13 02 31 21 53 343 122 321 X 21 21 3 2 100 010 001 343 122 321 rr rr 103 012 001 620 520 321 11-1- 012- 011- 1-00 5-2-0 2-01 32 12 rr rr 111 563 231 100 02-0 001 23 13 3 5 2 )1( rr rr r 111 2/533/2- 231 100 010 001 ) 2 1 ( 2r , 111 2/532/3 231 343 122 321 1 , 31 52 21 53 1 , 1313 0 0 2 2 202 32132 0303 012012 0000 6 1 00 0 02-020 0010 1 1 21 53 13 02 31 343 122 321 X 31 52 13 02 31 111 2/532/3 231 X 62 62 21 31 52 20 20 11 8.AAB=A+2B, 321 011 330 A B. AB=A+2B,(A- 2E)B=A, 121 011 332 2EA, 110110 01001-1 021310 100 010 001 121 011 332 32 12 2 rr rr 11-1-2-0 021310 031301 21 31 21 - rr rr rr 1/2-1/21/210 3/21/21/2-010 3/23/21/2-001 23 13 3 3 3- ) 2 1 ( rr rr r 111 311 331 2 1 )2( 1 EA , , 0 1-1 1 31010 1 3 1 r3 3131 111 311 331 2 1 )2( 1 AEAB 321 011 330 011 321 330 11.P- 1AP = , 20 01 , 11 41 P A11. 1 11 11111111 )()()()(PPPPPPPPPPA - 111 111 11 41 20 01 11 41 PP ) 11 41 3 1 ( 20 01 11 41 11 11 41 21 21 3 1 11 13 1111 1313 2421 2421 3 1 684683 27322731 3.3(A) 5.A b a A 1611 1723 14612 03211 1111 1 2020 0 0 1 1 3 3 1 1 ( ( 0 0 1111 1 P - 1- 1 111 4 ) ) 111 4 4 111 4 4 1313 24 4 242 b a A rr rr rr 4420 12610 12210 03211 31 21 41 3 2 20000 00800 12210 03211 42 32 2 b a rr rr (1)a=- 8,b=- 2,R(A)=2; (2)a=- 8,b- 2,R(A)=3; (3)a- 8,b=- 2,R(A)=3; (4)a- 8,b- 2,R(A)=4; 4.1(A) 2., , 02 , 0 , 0 321 321 321 xxx xxx xxx )1( 120 10 111 121 11 11 A =1=0, A=0, 3. , , , 1 2 321 321 321 xxx xxx xxx 4.1(A) 4.1(A) , 0 0 , 0, 0 , 0 3 3 3 x x 1)2)3) )2()1(23 11 11 11 23 A 1- 2, A0,. =- 2, 9330 6330 4211 4211 2121 1112 ),( 23 13 31 2 rr rr rr bA 3000 6330 4211 32 rr R(A)=2,R(A,b)=3, ; =1, 0000 0000 1111 1111 1111 1111 ),(bA R(A)=R(A,b)=13, ; 2 22 2 110 1110 11 11 11 111 ),( 12 32 21 rr rr rr bA =1, R(A)=R(A,b)=13, ; 1, 2 )1( )1(200 110 11 110 1110 11 ),( 32 23 rr rr bA 1,- 2, R(A)=R(A,b)=3,; =- 2, R(A)=2,R(A,b)=3,; 0000 000000 11111 000 0 0 1010 1 1 212 r1 rrr2 1 4.2(A) 2. ,5)(2)(3 311 ,)3 , 1 , 5 , 2( T 1 ,)10, 5 , 1 ,10( T 2 ,)1 , 1, 1 , 4( T 3 . 321 5)(2)(3 321 5236 T )3 , 1 , 5 , 2(3 T )10, 5 , 1 ,10(2,)1 , 1, 1 , 4(5 T T )24,18,12, 6( .)4 , 3 , 2 , 1( T 4. ,)2 , 1 , 1( T 1 ,)1 , 3( T 2 t T 3 ), 2 , 0(t t ,)2 , 1 , 1( T 1 ,)1 , 3( T 2 t T 3 ), 2 , 0(t 0 332211 xxx 0103 52 231 001 12 21 031 2 tt t t t tA . 25 tt 9. , 11 , 212 ,)2 , 1 , 1,)2 , 1 , T T t t 212 1 , 3(1 , 3( 2 ,) , 2 2 x x2 1 10 , 21rrr , 21 r , 21 0 2211rr xxx 0)()( 2121211rr xxx 0)()( 232121rrrr xxxxxxx r , 21 , 0 , 0 , 0 32 21 r r r x xxx xxx , 0 , 0 , 0 2 1 r x x x r , 21 4.3(A) 2. (2) 14011 31302 15120 12211 : x , 2 2 , 0 4.3(A) 4 3 2 1 14011 31302 15120 12211 A 1311 4152 0312 1021 1201 T A 2110 2550 2110 0220 1201 51 41 31 21 2 2 rr rr rr rr 2000 2000 2000 0110 1201 52 42 32 2 5 0.5 rr rr rr r 0000 0000 2000 0110 1201 53 43 rr rr R(A)=3,A ., 421 3. . 8 2 4 2 , 4 10 100 , 4 1 2 1 )1( 321 a 0 0 202 0 03 r3 1 2 2 2 0 0 1010 2020 5 r5 )=3, , 1 1, 211 0110 0960 010 844 2101 41002 21 ),( 14 4 34 24 4 1 2 321 aa rr r rr rr 000 000 010 211 12 2 43 32 )1( 96 1 11 rra r rr rr 2, ., 21 4. . 2 1 8 5 , 7 4 1 2 , 0 3 1 0 , 6 3 5 2 , 3 0 2 1 54321 2 27063 14330 81152 52021 ),( 54321 1313000 14330 23110 52021 41 21 3 2 rr rr 1313000 55000 23110 98201 12 2 32 2 )1( 3 rr r rr 00000 11000 10110 10201 13 3 43 23 8 ) 5 1 ( 13 3 rr r rr rr 3, , 421 ,2 213 . 4215 4.4(A) 1., 00000 0000 0 0101 ) ) 5 5 1 1 ( 1313 5 2 2 0 0000 11 1 02 2 4 r4 (1) , 097154 , 0347 , 0532 , 0253 4321 4321 4321 4321 xxxx xxxx xxxx xxxx : 97154 3471 1532 3681 97154 3471 1532 2153 A 12 rr 2131470 0210 717190 3681 41 31 21 4 2 rr rr rr 216300 0210 72100 31001 43 23 13 3 47 19 8 )1( rr rr rr r 0000 1300 0210 31001 32 2 42 7 1 3 rr r rr 0000 1300 0210 0101 13 3rr 03 , 0 2 , 0 43 32 31 xx xx xx ),( , 3 kx 23121 0 0 7 7 3 23 3 3 4747 1919 8 8 r3 rr r 23 3 r3 1 0 0 0000 10010 )1()1 3 3 020 3 3 , 3 1 2 1 4 3 2 1 k x x x x . 3 1 2 1 5. 3, 1,2,3 , 4 3 2 1 , 5 4 3 2 321 . : 3, 4- 3=1, 1,2,3 , 6 5 4 3 )(2)()( 3213121 4 4 . , )( , 1 kkx )( , 5 4 3 2 6 5 4 3 kkx 6., , 32235 , 12 2 , 5 )1( 4321 4321 21 xxxx xxxx xx 5 5 32235 12112 50011 ),(bA 222220 92110 50011 31 21 5 2 rr rr 42000 92110 42101 32 12 2 2 )1( rr rr r 21000 130110 80101 23 13 2 ) 2 1 ( rr rr r , 2 ,13 , 8 4 32 31 x xx xx ),( , 3 kx , 2 0 13 8 0 1 1 1 2 13 8 4 3 2 1 k k k k x x x x , 2 0 13 8 0 2 2 ), 1 1 1 1 k k . 0 1 1 1 5.1(A) 3.(x)1, 0,- 1,1(),(x) . : xxxx xxxxf 234 2 )1)(1()( 5.2(A) 1., 002 020 200 )1A , 013 211 233 )3A :1) )2()2( 02 020 20 2 AE xxxx x 2323 xxxx 2 )1)1x x x x A 1=22=- 2 1=2 (2E- A)x=0 , 1 0 1 0 1 0 21 ,A 1=2 212211 ,kkkk . 2=- 2 (- 2E- A)x=0 , 1- 0 1 3 ,A 2= 2 333 kk . 3) 13 211 233 AE 13 044 233 02 011 20 )4( 13 011 233 )4( (- 2(- 2 )4)(4( 2 A=4 =4(4E- A)x=0 , 1- 1 1 ,A=4 kk . 3.AAT : T T AEAEAE AAT 8. 1 1 1 221 102 11a A a : A AT T E E E E E E , 3 3 2 1 1 1 221 102 11aa A , 1 1 1 A . 3 , 1a 5.3(A) 2.:. :AB,6A B,. 4. 8102 331 6100 A , AA10. : 5.3(A)5.3(A) A A 5.3(A)5.3(A) B B 10 0 2420 331 610 8102 331 610 AE 120 031 02 )2( 120 331 610 )2( )1()2()23)(2( 22 A 1=12=2 1=1 (E- A)x=0 , 2 1 2 1 2=2 (2E- A)x=0 , 1 0 3