haihongyuan.com

# 同济大学线性代数第五版课后习题答案

1? 利用对角线法则计算下列三阶行列式?

201 (1)1?4?? ?183

201 解 1?4? ?183

?2?(?4)?3?0?(?1)?(?1)?1?1?8 ?0?1?3?2?(?1)?8?1?(?4)?(?1) ??24?8?16?4??4?

abc (2)bca? cab

abc 解 bca cab

?acb?bac?cba?bbb?aaa?ccc ?3abc?a3?b3?c3?

111 (3)abc? a2b2c2

111 解 abc a2b2c2

?bc2?ca2?ab2?ac2?ba2?cb2

?(a?b)(b?c)(c?a)?

xyx?y (4)yx?yx?

x?yxy

xyx?y 解 yx?yx

x?yxy

?x(x?y)y?yx(x?y)?(x?y)yx?y3?(x?y)3?x3

?3xy(x?y)?y3?3x2 y?x3?y3?x3

??2(x3?y3)?

2? 按自然数从小到大为标准次序? 求下列各排列的逆序数?

(1)1 2 3 4?

(2)4 1 3 2?

(3)3 4 2 1?

(4)2 4 1 3?

(5)1 3 ? ? ? (2n?1) 2 4 ? ? ? (2n)?

n(n?1) 解 逆序数为? 2

3 2 (1个)

5 2? 5 4(2个)

7 2? 7 4? 7 6(3个)

? ? ? ? ? ?

(2n?1)2? (2n?1)4? (2n?1)6? ? ? ?? (2n?1)(2n?2) (n?1个)

(6)1 3 ? ? ? (2n?1) (2n) (2n?2) ? ? ? 2?

3 2(1个)

5 2? 5 4 (2个)

? ? ? ? ? ?

(2n?1)2? (2n?1)4? (2n?1)6? ? ? ?? (2n?1)(2n?2) (n?1个) 4 2(1个)

6 2? 6 4(2个)

? ? ? ? ? ?

(2n)2? (2n)4? (2n)6? ? ? ?? (2n)(2n?2) (n?1个)

3? 写出四阶行列式中含有因子a11a23的项?

(?1)ta11a23a3ra4s?

(?1)ta11a23a32a44?(?1)1a11a23a32a44??a11a23a32a44? (?1)ta11a23a34a42?(?1)2a11a23a34a42?a11a23a34a42? 4? 计算下列各行列式?

41 (1)012512021

12542? 072024c2?c342??????1041 解 ?123202?104?1?102?122?(?1)4?3 ?140117c4?7c300103?14 ?4?110c2?c39910

123?142??????c00?2?0?

1?2c31714

2

(2)31?1141

522

031

62?

2

240 解 31?1211?c?4???c3?223?11402r4?r22521

06221522

0360?????3?14

22212

132

40

?r?4???r12?31?1142

022

030?0?

00

(3)?bdabacae

bf?cfcd?deef? 解 ?bdab

bf?ac

cfcdae

b?c

cce

?ee

??1111

1?11?1?4abcdef?

a1 (4)?001b?1001c?100? 1d

a1 解 ?001b?1001c?10r1?ar201?ab0??????1b10?1d00a1c?100 1d

a2abb2

(1)2aa?b2b?(a?b)3; 111

a2abb2c2?c1a2ab?a2b2?a2 2aa?b2b?????2ab?a2b?2a 00111c3?c11

222ab?ab?aab?a?(a?b)3 ? ?(b?a)(b?a)1 ?(?1)2b?a2b?2a3?1

ax?byay?bzaz?bxxyz (2)ay?bzaz?bxax?by?(a3?b3)yzx; az?bxax?byay?bzzxy 证明

ax?byay?bzaz?bx ay?bzaz?bxax?by

az?bxax?byay?bz

xay?bzaz?bxyay?bzaz?bx ?ayaz?bxax?by?bzaz?bxax?by zax?byay?bzxax?byay?bzxay?bzzyzaz?bx ?a2yaz?bxx?b2zxax?by zax?byyxyay?bz

xyzyzx ?a3yzx?b3zxy

zxyxyz

xyzxyz ?a3yzx?b3yzx

zxyzxy

xyz ?(a3?b3)yzx?

zxy

a2

2b (3)2c

d2

a2

2b 2c

d2(a?1)2(b?1)2(c?1)2(d?1)2(a?2)2(b?2)2(c?2)2(d?2)2(a?3)2(b?3)2(c?c? c?c? c?c得) (c?3)2433221(d?3)2

a2

2b ?c2

d22a?12b?12c?12d?12a?32b?32c?32d?32a?52b?5(c?c? c?c得) 2c?54332

2d?5

a2

2b ?c2

d2

1a (4)a2

a41bb2b42a?12b?12c?12d?11cc2c41d d2d4222222?0? 22

?(a?b)(a?c)(a?d)(b?c)(b?d)(c?d)(a?b?c?d);

1a a2

a41bb2b41cc2c41d d2d4

0b2(b2?a2)c2(c2?a2)d2(d2?a2)

111cd ?(b?a)(c?a)(d?a)b 222(b?a)c(c?a)d(d?a)

11c?bd?b ?(b?a)(c?a)(d?a)0 0c(c?b)(c?b?a)d(d?b)(d?b?a)

1 ?(b?a)(c?a)(d?a)(c?b)(d?b)(c?1b?a)d(d?b?a)

=(a?b)(a?c)(a?d)(b?c)(b?d)(c?d)(a?b?c?d)?

x0 (5) ? ?0an?1x? ? ?0an?10?1? ? ?0an?2? ? ?? ? ?? ? ?? ? ?? ? ?0000?? ? ??xn?a1xn?1? ? ? ? ?an?1x?an ? x?1a2x?a1

x?1?x2?ax?a? 命题成立? 当n?2时? D2?a122x?a1

?1

Dn?xDn?1?an(?1)n?1 ? x? ? 10?1 ? ? ? 1 ? ? ? ? ? ? ? ? ? ? ? ? 00 ? ? ? x00 ? ? ? ?1 ?xD n?1?an?xn?a1xn?1? ? ? ? ?an?1x?an ? 因此? 对于n阶行列式命题成立?

6? 设n阶行列式D?det(aij), 把D上下翻转、或逆时针旋转90?、或依副对角线翻转? 依次得

an1? ? ?anna1n? ? ?annann? ? ?a1n D1?? ? ?? ? ?? ? ?? D2?? ? ?? ? ?? ? ? ? D3?? ? ?? ? ?? ? ?? a11? ? ?a1na11? ? ?an1an1? ? ?a11证明D1?D2?(?1)n(n?1)

2D? D3?D ?

a11an1? ? ?ann D1?? ? ?? ? ?? ? ??(?1)n?1an1? ? ?a11? ? ?a1na21? ? ?? ? ?? ? ?? ? ?a1nann ? ? ?a2n

a11a21 ?(?1)n?1(?1)n?2an1? ? ?

a31? ? ?? ? ?? ? ?? ? ?? ? ?a1na2nann? ? ? ? ? ? ?a3n

n(n?1)

2 ?(?1)1?2?? ? ??(n?2)?(n?1)D?(?1)

D2?(?1) D3?(?1) n(n?1)112D? a? ? ?an1n(n?1)n(n?1)T? ? ?? ? ?? ? ??(?1)2D?(?1)2D? a1n? ? ?annn(n?1)2n(n?1)2D2?(?1)(?1)n(n?1)

2D?(?1)n(n?1)D?D?

7? 计算下列各行列式(Dk为k阶行列式)?

(1)Dn?

a0

Dn?0? ? ?010a0? ? ?0000a? ? ?00

00a? ? ?0? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?000? ? ?a0000? ? ?a100(按第n行展开) ? ? ?0a1a0?(?1)2n?a? ? ? 0? ? ?a(n?1)?(n?1)0(n?1)?(n?1)0an?1 ?(?1)0 ? ?0000? ? ?0

a

n?1n?(?1)?(?1) ? ? ?a(n?2)(n?2)?an?an?an?2?an?2(a2?1)?

x

(2)Dn? a? ?aax? ? ?a? ? ?? ? ?? ? ?? ? ?aa; ? ? ?x

? ? ?a? ? ?0? ? ?0? ? ? ?? ? ?0x?a 解 将第一行乘(?1)分别加到其余各行? 得 xaaa?xx?a0 Dn?a?x0x?a? ? ?? ? ?? ? ?a?x00

x?(n?1)aaa0x?a0 Dn?00x?a? ? ?? ? ?? ? ?000? ? ?a? ? ?0n?1? ? ?0?[x?(n?1)a](x?a)? ? ? ?? ? ?0x?aan(a?1)n

an?1(a?1)n?1

(3)Dn?1?? ? ?? ? ?aa?1

11? ? ?(a?n)n? ? ?(a?n)n?1; ? ? ?? ? ?? ? ?a?n? ? ?1

11a?1n(n?1)a Dn?1?(?1) ? ? ? ? ? ?an?1(a?1)n?1

an(a?1)n? ? ?1? ? ?a?n? ? ? ? ? ? n?1? ? ?(a?n)? ? ?(a?n)n

Dn?1?(?1) ?(?1) ?(?1) ?

an? ? ?

?? ? n(n?1)n?1?i?j?1n(n?1)2?[(a?i?1)?(a?j?1)] n?1?i?j?1n(n?1)2?[?(i?j)] n?(n?1)?? ? ??12?(?1)?n?1?i?j?1?(i?j) n?1?i?j?1?(i?j)? ?? ? ? ? ?bn (4)D2n?cna1b1c1d1; dn

bn 解 an? ? ?? ? ?? ? ?? ? ? D2n?cna1b1c1d1(按第1行展开) dnan?1

?an? ? ??? ? a1b1c1d1

??? ? ? ? ?bn?10 cn?10dn?100dn

0an?1? ? ?

? ? ? ?(?1)2n?1bn

cn?1

cna1b1c1d1? ? ?? ? ?bn?1? dn?10

i?2n

n

i?1所以 D2n??(aidi?bici)?

(5) D?det(aij)? 其中aij?|i?j|;

01

Dn?det(aij)?23? ? ?n?1

?1r1?r2?11 ???????1r2?r3? ? ? ? ? ? n?1123012101210? ? ?? ? ?? ? ?n?2n?3n?411? ? ?11? ? ??11? ? ??1?1? ? ?? ? ?? ? ?? ? ?n?3n?4? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?111 1? ? ?0n?1n?2n?3 n?4? ? ?01?1?1?1? ? ?n?2

?1c2?c1?1

1 ???????1c3?c1? ? ?

? ? ? n?1000?200?2?20?2?2?2? ? ?? ? ?? ? ?2n?32n?42n?5? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?000 0? ? ?n? ?(?1)n?1(n?1)2n?2?

?a11

(6)Dn?11?a2

? ? ?? ? ?11 解

?a11

Dn?11?a2

? ? ?? ? ?11

? ? ?1

? ? ?1, 其中aa ? ? ? a?0?

12n

? ? ?? ? ?? ? ?1?an

? ? ?1? ? ?1 ? ? ?? ? ?? ? ?1?an

a1

c1?c2?a2

?????0

c2?c3? ? ?

0 ? ? ?

00a2?a3? ? ?0000a3? ? ?00? ? ?0? ? ?0? ? ?0? ? ?? ? ?? ? ??an?1? ? ?0

010101 ? ? ?? ? ?an?11?an1?an

1?1

?a1a2? ? ?an0

? ? ?0001?1? ? ?00001? ? ?00? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?000? ? ??10

0a1?1

?1

0a2

?1

0a3

? ? ?? ? ?

?1

1an?1

?1

?11?an

100

?a1a2? ? ?an ? ?

0010? ? ?0001? ? ?0? ? ?? ? ?? ? ?? ? ?? ? ?000? ? ?0000? ? ?1

a1?1?1a2?1a3? ? ??1an?1

ni?1

000? ? ?001??ai?1

?(a1a2?an)(1??1)?

i?1ai

8? 用克莱姆法则解下列方程组? ?x1?x2?x3?x4?5?x?2x2?x3?4x4??2

(1)?1?

2x1?3x2?x3?5x4??2?3x?x?2x?11x?0?1234

n

D?12

3

12?31

1?1?12

1

4??142? ?511

52 D1???20 D3?2

3所以 x1?

12?311?1?1214??142? D?2?52

1135

?2?201?1?121

4??284? ?511

12?315?2?20114??426? D?1

42?5

1131

2?311?1?125

?2?142? ?20

DDDD?1? x2??2? x3??3? x4???1?

DDDD

?1?5x1?6x2?0??x1?5x2?6x3 (2)?x2?5x3?6x4?0? x3?5x4?6x5?0??x4?5x5?1?

51 D?000651000651000651000?665? 65

0 D1?0051 D3?00065100651000651010001006510065105010?1507? D2?0605005010?703? D4?06050100016510006510065100065110001000??1145? 65000??395? 655 D5?000651000651000651100?212? 01

x1?1507? x2??1145? x3?703? x4??395? x4?212? 665665665665665

???x1?x2?x3?0 9? 问?? ?取何值时? 齐次线性方程组?x1??x2?x3?0有非??x1?2?x2?x3?0

?1 D?1??????? 12? 令D?0? 得

??0或??1?

??(1??)x1?2x2?4x3?0 10? 问?取何值时? 齐次线性方程组?2x1?(3??)x2?x3?0??x1?x2?(1??)x3?0

???24???3??4 D?23??1?21??1 111??101??

?(1??)3?(??3)?4(1??)?2(1??)(?3??)

?(1??)3?2(1??)2???3?

??0? ??2或??3?

1? 已知线性变换?

??x1?2y1?2y2?y3?x2?3y1?y2?5y3? ??x3?3y1?2y2?3y3

?x1??221??y1? ?x2???315??y2?? ?x??323??y???2??3??

?y1??221??x1???7?49??y1?故 ?y2???315??x2???63?7??y2?? ?y??323??x??32?4?????3????y3??2???1

??y1??7x1?4x2?9x3 ?y2?6x1?3x2?7x3? ??y3?3x1?2x2?4x3 2? 已知两个线性变换

??x1?2y1?y3 ?x2??2y1?3y2?2y3? ??x3?4y1?y2?5y3 解 由已知 ??y1??3z1?z2?y2?2z1?z3? ??y3??z2?3z3求从z1? z2? z3到x1? x2? x3的线性变换?

?x1??201??y1??201???31 ?x2????232??y2????232??20?x??415??y??415??0?1??2?????3??

??613??z1? ??12?49??z2?? ??10?116??z????3?0??z1?1??z2? ??3???z3?

??x1??6z1?z2?3z3所以有?x2?12z1?4z2?9z3? ??x3??10z1?z2?16z3

?111??123? 3? 设A??11?1?? B???1?24?? 求3AB?2A及ATB? ?1?11??051?????

?111??123??111? 解 3AB?2A?3?11?1???1?24??2?11?1? ?1?11??051??1?11???????

?058??111???21322? ?3?0?56??2?11?1????2?1720?? ?290??1?11??429?2????????111??123??058? ATB??11?1???1?24???0?56?? ?1?11??051??290???????

4? 计算下列乘积?

?431??7? (1)?1?23??2?? ?570??1?????

?431??7??4?7?3?2?1?1??35? 解 ?1?23??2???1?7?(?2)?2?3?1???6?? ?570??1??5?7?7?2?0?1??49?????????

?3? (2)(123)?2?? ?1???

?3? 解 (123)?2??(1?3?2?2?3?1)?(10)? ?1???

?2? (3)?1?(?12)? ?3???

?2?(?1)2?2???2?2? 解 ?1?(?12)??1?(?1)1?2????1?3??3?(?1)3?2???3?????

?1?02140?? (4)???11?134????43?1?301?2? ? 1?

?2??4?2?? 6??

?1?02140?? 解 ???11?134????43?1?301?2??6?78?? ?1???20?5?6????2?

?a11a12a13??x1? (5)(x1x2x3)?a12a22a23??x2?? ????aaa?132333??x3?

?a11a12a13??x1? (x1x2x3)?a12a22a23??x2? ????aaa?132333??x3?

?x1? ?(a11x1?a12x2?a13x3 a12x1?a22x2?a23x3 a13x1?a23x2?a33x3)?x2? ?x??3?

22 ?a11x12?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3?

1 5? 设A???1?2?? B??10?? 问? ?12?3????

(1)AB?BA吗?

3 因为AB???4?4?? BA??12?? 所以AB?BA? ?38?6????

(2)(A?B)2?A2?2AB?B2吗? 解 (A?B)2?A2?2AB?B2?

2 因为A?B???2?2?? 5??

2 (A?B)2???2?2??2?25???2???814?? ?1429?5????

38??68???10???1016?? 但 A2?2AB?B2???411???812??34??1527?????????

(3)(A?B)(A?B)?A2?B2吗? 解 (A?B)(A?B)?A2?B2?

2 因为A?B???2?2?? A?B??0?05???

2??0?05???2?? 1??6?? 9??2 (A?B)(A?B)???2?2???0?01???

38???10???2而 A2?B2???411??34??1?????

6? 举反列说明下列命题是错误的?

(1)若A2?0? 则A?0?

0 解 取A???0?1?? 则A2?0? 但A?0? 0??

(2)若A2?A? 则A?0或A?E?

1 解 取A???0?

1 A???0?0?? X??11?? Y??1??11??00?????1?? 1??

10?? 求A2? A3? ? ? ?? Ak? 7? 设A????1???

10?10???10?? 解 A2????1????1??2?1???????

10?10???10?? A3?A2A???2?1????1??3?1???????

? ? ? ? ? ??

1 Ak???k??0?? 1??

??10? 8? 设A??0?1?? 求Ak ? ?00????

??10???10???22?1? A2??0?1??0?1???0?22??? ?00???00???00?2???????

??33?23?? A3?A2?A??0?33?2?? ?00?3???

??44?36?2? A4?A3?A??0?44?3?? ?00?4???

??55?410?3? A5?A4?A??0?55?4?? ?00?5???

? ? ? ? ? ??

??kk?k?1k(k?1)?k?2?2k A??0?kk?k?1?00?k???? ? ??

??kk?k?1k(k?1)?k?2?????10?2 Ak?1?Ak?A??0?kk?k?1??0?1? ?00??00???k??????

??k?1(k?1)?k?1(k?1)k?k?1???2?k?1(k?1)?k?1?? ??0??k?100?????由数学归纳法原理知?

??kk?k?1k(k?1)?k?2???2k?k?1?? Ak??0?k?00??k????

9? 设A? B为n阶矩阵，且A为对称矩阵，证明BTAB也是对称矩阵?

(BTAB)T?BT(BTA)T?BTATB?BTAB?

10? 设A? B都是n阶对称矩阵，证明AB是对称矩阵的充分必要条件是AB?BA?

AB?(AB)T?BTAT?BA?

11? 求下列矩阵的逆矩阵?

1 (1)??2?2?? 5??

12?? |A|?1? 故A?1存在? 因为 解 A???25???

A11A21??5?2? A*???AA????21?? ??1222??

5?2? 故 A?1?1A*????21??|A|??

cos??sin?? (2)??sin?cos?????

co?s?si?n?? |A|?1?0? 故A?1存在? 因为 解 A???si?s??nco??

A11A21??cos?sin?? A*???AA????sin?cos??? ??1222??

cos?sin??? 所以 A?1?1A*????sin?cos??|A|??

?12?1? (3)?34?2?? ?5?41???

?12?1? 解 A??34?2?? |A|?2?0? 故A?1存在? 因为 ?5?41????A11A21A31???420? A*??A12A22A32????136?1?? ????3214?2?AAA??132333??

??210??13?11?1所以 A?A*???3??? 22?|A|??167?1??

?a1a?0??2 (4)??(a1a2? ? ?an ?0) ?

???0an??

?a1?0?a?2 解 A???? 由对角矩阵的性质知 ??0?a?n??1??a1?0?1a??12?? A????1?0????a?n?

12? 解下列矩阵方程?

2 (1)??1?5?X??4?6?? ?21?3????

5??4?6??3?5??4?6??2?23?? ??????12 解 X????13??21????12????2

(2)X??21?1?1?

?210???13?1?11??

??432????

?1

?432???2

?211?01??

?1?11??

?1?1?13??10

3??432?????23?12??

??330??

????221?

???

??8

35?2

3??

(3)???1142?01???31?

???X??2

??1????0?1???

?1

??12????30?11???20

??????11???

?1?2?4?31??10?12??11?????0?1????12??

?1??660???102?12?3???1?????11?

?1??

?40??

(4)??010??100??

?100?X?001???12?04?31???001????010??

???1?20??1????08??

?010??1?43??100? 解 X??100??20?1??001? ?001??1?20??010???????

?010??1?43??100??2?10??1?1 ???100??20?1??001???

?001????1?20?1

???010????1

13? 利用逆矩阵解下列线性方程组?

x?2x

(1)??12?3x3?1

?2x1?2x2?5x3?2?

??3x1?5x2?x3?3

??1

?223??x1??1?

?3255??

1???x2???2??

?x???3?

3??

1??123??1??1

?x???225?

???2???0?

2??

x??

3??351???3????0??

?x2?0?

??x3?0

(2)??x1?x

?x2?x3?2

?2x12?3x

x3?1?

??3x1?2x2?53?0

??1?1?1??x1?

?2?1?3??x?2?

2???

?32?5???1??

?x??

3??0??30??4?2?? ?

?x1??1?1?1??2??5?故 ?x2???2?1?3??1???0?? ?x??32?5??0??3???????3???1

x?5??1故有 ?x2?0?

??x3?3

14? 设Ak?O (k为正整数)? 证明(E?A)?1?E?A?A2?? ? ??Ak?1? 证明 因为Ak?O ? 所以E?Ak?E? 又因为

E?Ak?(E?A)(E?A?A2?? ? ??Ak?1)?

(E?A)?1?E?A?A2?? ? ??Ak?1?

E?(E?A)?(A?A2)?A2?? ? ??Ak?1?(Ak?1?Ak)

?(E?A?A2?? ? ??A k?1)(E?A)?

(E?A)?1(E?A)?E?A?A2?? ? ??Ak?1?

15? 设方阵A满足A2?A?2E?O? 证明A及A?2E都可逆? 并求A?1及(A?2E)?1?

A2?A?2E? 即A(A?E)?2E?

A2?A?6E??4E? 即(A?2E)(A?3E)??4E?

?A?1A(A?E)?2A?1E?A?1?1(A?E)? 2

? (A?2E)(A?3E)??4 E?

(A?2E)?1?1(3E?A)? 4

16? 设A为3阶矩阵? |A|?1? 求|(2A)?1?5A*|? 2

|(2A)?1?5A*|?|1A?1?5|A|A?1|?|1A?1?5A?1| 222

?|?2A?1|?(?2)3|A?1|??8|A|?1??8?2??16? 17? 设矩阵A可逆? 证明其伴随阵A*也可逆? 且(A*)?1?(A?1)*?

|A*|?|A|n|A?1|?|A|n?1?0?

(A*)?1?|A|?1A? ?1?1又A?1(A)*?|A|(A)*? 所以 ?1|A|

(A*)?1?|A|?1A?|A|?1|A|(A?1)*?(A?1)*?

18? 设n阶矩阵A的伴随矩阵为A*? 证明?

(1)若|A|?0? 则|A*|?0?

(2)|A*|?|A|n?1?

(1)用反证法证明? 假设|A*|?0? 则有A*(A*)?1?E? 由此得 A?A A*(A*)?1?|A|E(A*)?1?O ?

(2)由于A?1?1A*? 则AA*?|A|E? 取行列式得到 |A|

|A||A*|?|A|n?

?033? 19? 设A??110?? AB?A?2B? 求B? ??123???

?1??233??033??033? B?(A?2E)?1A??1?10??110????123?? ??121???123??110???????

?101? 20? 设A??020?? 且AB?E?A2?B? 求B? ?101???

(A?E)B?A2?E?

001 因为|A?E|?010??1?0? 所以(A?E)可逆? 从而 100

?201? B?A?E??030?? ?102???

21? 设A?diag(1? ?2? 1)? A*BA?2BA?8E? 求B?

??8(|A|E?2A)?1

??8(?2E?2A)?1

?4(E?A)?1

?4[diag(2? ?1? 2)]?1 ?4diag(1, ?1, 1) 22

?2diag(1? ?2? 1)?

?1?0 22? 已知矩阵A的伴随阵A*??1?0?010?300100?0?? 0?

8??

B?3(A?E)?1A?3[A(E?A?1)]?1A ?3(E?1A*)?1?6(2E?A*)?1 2

?1?0 ?6??1?0?010300100??600???060??60?03?6????100600?0?? 0?

?1??

?1?4? ????1 23? 设P?1AP??? 其中P???11???0???

1 |P|?3? P*????1?

?1而 ?11???0?110?? 求A11? 2?? 解 由P?1AP??? 得A?P?P?1? 所以A11? A=P?11P?1. 4?? P?1?1?14?? ??1?3??1?1??0????10 ?? ?0211?2????

?14????27312732??1?4?10????1133故 A????0211??11????683?684?? 11???????????33?

??1??111? 24? 设AP?P?? 其中P??10?2?? ???1?? ?1?11???5????

?diag(1?1?58)[diag(5?5?5)?diag(?6?6?30)?diag(1?1?25)] ?diag(1?1?58)diag(12?0?0)?12diag(1?0?0)?

?(A)?P?(?)P?1

?1P?(?)P* |P|

?111??100???2?2?2? ??2?10?2??000???303??1?11??000???12?1???????

?111? ?4?111?? ?111???

25? 设矩阵A、B及A?B都可逆? 证明A?1?B?1也可逆? 并

A?1(A?B)B?1?B?1?A?1?A?1?B?1? 而A?1(A?B)B?1是三个可逆矩阵的乘积? 所以A?1(A?B)B?1可逆? 即A?1?B?1可逆?

(A?1?B?1)?1?[A?1(A?B)B?1]?1?B(A?B)?1A?

?1?0 26? 计算?0?0?210010200??11??01??0?03???031?12?1?? 0?23?00?3??

1 解 设A1???0?2?? A??2

2?01???1?? B??31?? B???23?? 1?2?1?2?0?3?3??????

A1E??EB1??A1A1B1?B2?则 ??OA??OB???OAB?? ??2??2?22?

1?而 AB?B??112?0

2 A2B2???0?2??31????23???52?? ?2?1??0?3??2?4?1????????1???23????43?? ?0?3??0?9?3??????

252?12?4?? 0?43?

00?9???1A1E??EB1??A1A1B1?B2??0???所以 ???????0OAOBOAB??2??2?22??0?

?1?0即 ?0?0?210010200??11??01??0?3???0031??112?1???00?23??0?00?3???0252?12?4?? 0?43?

00?9??

1 27? 取A?B??C?D???0?0?? 验证AB? |A||B|? 1?CD|C||D|?

1

34?? A??20?? 解 令A1???4?3?2?22?????

A1O?则 A???OA?? ?2?

8AOO??8?A故 A??1???18??? OAOA?2??2?8

888816 |A8|?|A|A1||A2|?1||A2|?10?

?540O?4?054?O?4?A1 A??? ?4???4?OA2??O2604?22??

29? 设n阶矩阵A及s阶矩阵B都可逆? 求

OA? (1)??BO?????1

OA???C1C2?? 则 解 设??BO??CC????34??1

OA??C1C2???AC3AC4???EnO?? ??BO??CC??BCBC??OE????34??1s?2???AC3?En?C3?A?1

??由此得 ?AC4?O??C4?O? BC?OC?O?C1?B?1?BC1?E?2s?2

?1OAOB??? ??所以 ?BO????1????AO??1

AO?? (2)??CB????1

AO??D1D2?? 则 解 设??CB???DD????34?

?1AOA??? ??所以 ?CB????1?1O?1??BCAB?????1

30? 求下列矩阵的逆阵?

?5?2 (1)?0?0?210000850?0?? 3?

2??

5 解 设A???2?2?? B??83?? 则 ?52?1????

5 A?1???2?2???1?2?? B?1??8??25??51?????

?1?13???2?3?? ??58?2?????1?5?2于是 ?02100080??1?200??1?10???A???A????2500?? ????1??03??02?3BB??????0052????00?5

?00

(2)?1

?1200??

?21030??

?1214??

??? B???3

?14?

???? C???212?

???? 则

?10?1

??1220000???1

?13

?1210???AOB?

???

4??C????A?1

??B?1CA?1BO?1???

??0?

??11100

0??

??220

??1?110?

?263??

?11?

?8?5

24?1

124??

1? 把下列矩阵化为行最简形矩阵? 8??

?102?1? (1)?2031?? ?304?3????102?1? 解 ?2031?(下一步? r2?(?2)r1? r3?(?3)r1? ) ?304?3????102?1? ~?00?13?(下一步? r2?(?1)? r3?(?2)? ) ?00?20????102?1? ~?001?3?(下一步? r3?r2? ) ?0010????102?1? ~?001?3?(下一步? r3?3? ) ?0003????102?1? ~?001?3?(下一步? r2?3r3? ) ?0001???

?102?1? ~?0010?(下一步? r1?(?2)r2? r1?r3? ) ?0001???

?1000? ~?0010?? ?0001???

?02?31? (2)?03?43?? ?04?7?1????02?31? 解 ?03?43?(下一步? r2?2?(?3)r1? r3?(?2)r1? ) ?04?7?1???

?02?31? ~?0013?(下一步? r3?r2? r1?3r2? ) ?00?1?3????02010? ~?0013?(下一步? r1?2? ) ?0000????0105? ~?0013?? ?0000????1?13?43??3?35?41? (3)?? 2?23?20??3?34?2?1????1?3 解 ?2?3?

?1?0 ~?0?0??1?3?2?33534?4?4?2?23?1?(下一步? r?3r? r?2r? r?3r? ) 2131410??1???13?43?0?48?8?(下一步? r?(?4)? r?(?3) ? r?(?5)? ) 2340?36?6?0?510?10???1?0 ~?0?0?

?1?0 ~?0?0??1000?10003111?4?2?2?23?2?(下一步? r?3r? r?r? r?r? ) 1232422?2??02?3?1?22?? 000?

000???23?12 (4)?3?2?2?3?1?3?7?0?2?4?? 830?743??

?23?12 解 ?3?2?2?3?1?3?7?0?2?4?(下一步? r?2r? r?3r? r?2r? ) 123242830?743??

?0?1111??120?2?4? ~?(下一步? r2?2r1? r3?8r1? r4?7r1? ) 0?88912??0?77811???

?0?1 ~?0?0?

?1?0 ~?0?0??10000100120011?0?2?(下一步? r?r? r?(?1)? r?r? ) 1224314?14??0?2??1?1?(下一步? r?r? ) 2314?

00??2?100

?1?0 ~?0?0?01002?1000010?2?3?? 4?

0??

?010??101??123? 2? 设?100?A?010???456?? 求A? ?001??001??789???????

?010? 解 ?100?是初等矩阵E(1? 2)? 其逆矩阵就是其本身? ?001???

?101? ?010?是初等矩阵E(1? 2(1))? 其逆矩阵是 ?001???

?10?1? E(1? 2(?1)) ??010?? ?001???

?010??123??10?1? A??100??456??010? ?001??789??001???????

?456??10?1??452?

??123??010???122?? ?789??001??782???????

3? 试利用矩阵的初等变换? 求下列方阵的逆矩阵?

?321? (1)?315?? ?323???

?321100??321100? 解 ?315010?~?0?14?110? ?323001??002?101?????

?3203/20?1/2??3007/22?9/2? ~?0?1011?2?~?0?1011?2? ?002?10??1????001?1/201/2?

?1007/62/3?3/2? ~?010?1?12? ?001?1/201/2???

?72?3??632?

?3?20?1??0221? (2)?? 1?2?3?2??0121???

?3?20?11000??02210100? 解 ? 1?2?3?20010??01210001????1?2?3?20010??01210001? ~? 049510?30??02210100????1?2?3?20010??01210001? ~? 001110?3?4??00?2?1010?2????1?2?3?20010??0121000?1 ~? 001110?3?4??000121?6?10????1?200?0100 ~?0010?0001?

?1?0 ~?0?0?01000010?10?12?11`?11?2?2?0?1? 36??6?10??

?1?0故逆矩阵为??1?2?011?2?4?0010?1? 0?1?136?121?6?10??1?2?4?10?1?? ?136?1?6?10??

?41?2??1?3? 4? (1)设A??221?? B??22?? 求X使AX?B? ?31?1??3?1?????

?41?21?3?r?100102? (A, B)??221 22?~ ?010 ?15?3?? ?31?13?1??001124??????102?所以 X?AB???15?3?? ?124????1

?021?123?? 求X使XA?B? (2)设A??2?13?? B???2?31???33?4?????

?02?312?r?1002?4? (AT, BT)??2?132?3?~ ?010?17?? ?13?431??001?14?????

?2?4?所以 XT?(AT)?1BT???17?? ??14???

2?1?1? 从而 X?BA?1????474????

?1?10? 5? 设A??01?1?? AX ?2X?A? 求X? ??101???

??1?101?10? (A?2E, A)??0?1?101?1? ??10?1?101???

?10001?1? ~?010?101?? ?0011?10???

?01?1?所以 X?(A?2E)?1A???101?? ?1?10???

6? 在秩是r 的矩阵中,有没有等于0的r?1阶子式? 有没有等于0的r阶子式?

?1000? 例如? A??0100?? R(A)?3? ?0010???

00000 是等于0的2阶子式? 100是等于0的3阶子式? 00010

7? 从矩阵A中划去一行得到矩阵B? 问A? B的秩的关系怎样?

8? 求作一个秩是4的方阵? 它的两个行向量是

(1? 0? 1? 0? 0)? (1? ?1? 0? 0? 0)?

?1?1?1?0?0?0?100000100000100?0?0?? 0?

0??

9? 求下列矩阵的秩? 并求一个最高阶非零子式?

?3102? (1)?1?12?1?; ?13?44???

?3102? 解 ?1?12?1?(下一步? r1?r2? ) ?13?44???

?1?12?1? ~?3102?(下一步? r2?3r1? r3?r1? ) ?13?44???

?1?12?1? ~?04?65?(下一步? r3?r2? ) ?04?65???

1?12?1? ~?04?65?? ??0000?

?32?1?3?2? 解 ?2?131?3?(下一步? r1?r2? r2?2r1? r3?7r1? ) ?705?1?8???13?4?41? ~?0?7119?5?(下一步? r?3r? ) ??0?213327?15?13?4?41? ~?0?7119?5?? ??00000?32

32矩阵的秩是2? ??7是一个最高阶非零子式? 2?1

?218?2?30 (3)?3?25?103?37?7?5?? 80?20??

37?7?5?(下一步? r?2r? r?2r? r?3r? ) 14243480?

20??

7??5?(下一步? r?3r? r?2r? ) 21310?

0???218?2?30 解 ?3?25?103??012?1?0?3?63 ~?0?2?42?1032?

?0?0 ~?0?1?

?0?0 ~?0?1?1000100020032003?17?016?(下一步? r?16r? r?16r? ) 2432014?20???10027?1? 0?

0??

?1?0 ~?0?0?010032002?1000?7?? 1?

0??

07?5矩阵的秩为3? 580?70?0是一个最高阶非零子式? 320

10? 设A、B都是m?n矩阵? 证明A~B的充分必要条件是R(A)?R(B)?

A~D? D~B?

?1?23k? 11? 设A???12k?3?? 问k为何值? 可使 ?k?23???

(1)R(A)?1? (2)R(A)?2? (3)R(A)?3?

k?1?23k?r?1?1??? 解 A???12k?3?~ ?0k?1k?1?k?23??00?(k?1)(k?2)?????

(1)当k?1时? R(A)?1?

(2)当k??2且k?1时? R(A)?2?

(3)当k?1且k??2时? R(A)?3?

12? 求解下列齐次线性方程组:

??x1?x2?2x3?x4?0 (1)?2x1?x2?x3?x4?0? ??2x1?2x2?x3?2x4?0

0??112?1??10?1 A??211?1?~?013?1?? ?2212??001?4/3?????

?x?4x?134

?x??3x4于是 ?2? 4?x3?x4?x?3?4x4

?4??x1??3??x???3? ?2??k?4?(k为任意常数)? ?x3????x4??3??1?

??x1?2x2?x3?x4?0 (2)?3x1?6x2?x3?3x4?0? ??5x1?10x2?x3?5x4?0

?121?1??120?1? A??36?1?3?~?0010?? ?5101?5??0000?????

?x1??2x2?x4?x2?x2于是 ?? x3?0?x?x?44

?x1???2??1??x??1??0? ?2??k1???k2??(k1? k2为任意常数)? 00?x3??0??1??x4?????

?2x1?3x2?x3?5x4?0?3x?x?2x3?7x4?0 (3)?12? 4x1?x2?3x3?6x4?0?x??12x2?4x3?7x4?0

?23?31 A??41?1?2??15??12?7?~?0?36??0?4?7???0010000100?0?? 0?

1??

?x1?0?x?0于是 ?2? x3?0?x?0?4

?x1?0?x?0 ?2? x?0?x3?0?4

?3x1?4x2?5x3?7x4?0?2x?3x2?3x3?2x4?0 (4)?1? 4x1?11x2?13x3?16x4?0?7x?2x?x?3x?0?1234

?34?5?2?33 A??411?13?7?21??17???2?~?016???03????00?3171?1917000013?17?20?? ?17?00?? ?x?3x?13x?1173174

?1920于是 ?x2?x3?x4? 1717?x?x?x3?x3

?44

?3???13??x1??17??17??x??19??20? ?2??k1???k2???(k1? k2为任意常数)? ?x3??17??17??x4??1??0??0??1?

13? 求解下列非齐次线性方程组:

??4x1?2x2?x3?2 (1)?3x1?1x2?2x3?10? ??11x1?3x2?8

?42?12??13?3?8? B??3?1210?~?0?101134?? ?11308?000?6????

?2x?3y?z?4?x?2y?4z??5 (2)?? 3x?8y?2z?13?4x?y?9z??6?

?231?1?24 B??38?2?4?19?4??1?5?~?013??0?0?6???01002?100?1?2?? 0?

0??

??x??2z?1于是 ?y?z?2? ??z?z

?x???2???1?即 ?y??k?1???2?(k为任意常数)? ?z??1??0???????

??2x?y?z?w?1 (3)?4x?2y?2z?w?2? ??2x?y?z?w?1

?21?111??11/2?1/201/2? B??42?212?~?00010?? ?21?1?11??00000?????

?x??1y?1z?1?222?于是 ?y?y?

?z?z??w?0

1??1??1???x???2??2??y??2?

??2x?y?z?w?1 (4)?3x?2y?z?3w?4? ??x?4y?3z?5w??2

?21?111??10?1/7?1/76/7? B??3?21?34?~?01?5/79/7?5/7?? ?14?35?2?00000????

?x?1z?1w?6?777?于是 ?y?5z?9w?5? 777?z?z?w?w?

?1??1??6??x??7??7??7??y??5??9??5?即 ???k1???k2???????(k1? k2为任意常数)? z7??7????w??7

???1??0??0??0??1??0?

14? 写出一个以

?2???2???3??4?x?c1???c2?? 10?0??1?????

?x1??2???2??x???3??4? ?2??c1???c2??? 10?x3??0??1??x4?????

?x1?2c1?c2?x??3c1?4c2 ?2? x3?c1?x?c?42

x1?2x3?x4或 ??x??3x?4x? ?234

x1?2x3?x4?0或 ??x?3x?4x?0? ?234

15? ?取何值时? 非齐次线性方程组

???x1?x2?x3?1

?x1??x2?x3???

2?x?x??x???123

(1)有唯一解? (2)无解? (3)有无穷多个解?

??111? 解 B??1?1?? ?11??2???

211??? ~ ?0??1? 1???(1??)???00(1??)(2??)(1??)(??1)2?r

(1)要使方程组有唯一解? 必须R(A)?3? 因此当??1且???2

(2)要使方程组无解? 必须R(A)?R(B)? 故

(1??)(2??)?0? (1??)(??1)2?0?

(3)要使方程组有有无穷多个解? 必须R(A)?R(B)?3? 故 (1??)(2??)?0? (1??)(??1)2?0?

16? 非齐次线性方程组

???2x1?x2?x3??2

?x1?2x2?x3??

2?x?x?2x???123

?????211?2??1?21?2???(??1)?? 解 B?1?21?~?01?1?11?2?2??3???000(??1)(??2)??

??211?2??10?11? B??1?211?~?01?10?? ?11?21??0000?????

x1?x3?1??x?x?1 ??x1?x3或?x2?x3? ?23??x3?x3

?x1??1??1?即 ?x2??k?1???0?(k为任意常数)? ?x??1??0??3?????

??211?2??10?12? B??1?21?2?~?01?12?? ?11?24??0000?????

x1?x3?2??x1?x3?2 ??x?x?2或?x2?x3?2? ?23??x3?x3

?x1??1??2?即 ?x2??k?1???2?(k为任意常数)? ?x??1??0??3?????

??(2??)x1?2x2?2x3?1 17? 设?2x1?(5??)x2?4x3?2? ???2x1?4x2?(5??)x3????1

2?21??2?? 解 B??25???42? ??2?45?????1???

?42?25?????? ~01??1??1???00(1??)(10??)(1??)(4??)???

(1??)(10??)?0且(1??)(4??)?0?

?12?21? B~?0000?? ?0000???

??x1??x2?x3?1 ?x2? x2? ??x3? x3

?x1???2??2??1?或 ?x2??k1?1??k2?0???0?(k1? k2为任意常数)? ?x??0??1??0??3???????

18? 证明R(A)?1的充分必要条件是存在非零列向量a及非零行向量bT? 使A?abT?

?1?0 ?????0?00???0????????????0??1?0???0?(1, 0, ???, 0)? ??????????0?0????

?1??1??0????10 PAQ???(1, 0, ???, 0)? 或A?P??(1, 0, ???, 0)Q?1? ???????0??0?????

?1???T?10 令a?P??? b?(1? 0? ???? 0)Q?1? 则a是非零列向量? bT是非????0???

1?R(A)?R(abT)?min{R(a)? R(bT)}?min{1? 1}?1? 所以R(A)?1?

19? 设A为m?n矩阵? 证明

(1)方程AX?Em有解的充分必要条件是R(A)?m? 证明 由定理7? 方程AX?Em有解的充分必要条件是

R(A)?R(A? Em)?

(2)方程YA?En有解的充分必要条件是R(A)?n?

20? 设A为m?n矩阵? 证明? 若AX?AY? 且R(A)?n? 则X?Y? 证明 由AX?AY? 得A(X?Y)?O? 因为R(A)?n? 由定理9? 方程A(X?Y)?O只有零解? 即X?Y?O? 也就是X?Y?

1? 设v1?(1? 1? 0)T? v2?(0? 1? 1)T? v3?(3? 4? 0)T? 求v1?v2及3v1?2v2?v3?

?(1?0? 1?1? 0?1)T ?(1? 0? ?1)T?

3v1?2v2?v3?3(1? 1? 0)T ?2(0? 1? 1)T ?(3? 4? 0)T

?(3?1?2?0?3? 3?1?2?1?4? 3?0?2?1?0)T ?(0? 1? 2)T?

2? 设3(a1?a)?2(a2?a)?5(a3?a)? 求a? 其中a1?(2? 5? 1? 3)T? a2?(10? 1? 5? 10)T? a3?(4? 1? ?1? 1)T?

a?1(3a1?2a2?5a3) 6

?1[3(2, 5, 1, 3)T?2(10, 1, 5, 10)T?5(4, 1, ?1, 1)T] 6

?(1? 2? 3? 4)T?

3? 已知向量组

A? a1?(0? 1? 2? 3)T? a2?(3? 0? 1? 2)T? a3?(2? 3? 0? 1)T? B? b1?(2? 1? 1? 2)T? b2?(0? ?2? 1? 1)T? b3?(4? 4? 1? 3)T? 证明B组能由A组线性表示? 但A组不能由B组线性表示? 证明 由

?0?1 (A, B)??2?3?

?1r? ~ ?00?0?30122301204??1r?1?24?~0 111??0?213???0031?24?32204? 1?6?15?7?2?8?17?9??031?24?1?6?15?7? 041?35?00000??031?6020041?24??1r??15?7?~0 5?1525??0?1?35???0

?204??102??1?1?24?r?0?22?r?0~~ B??111??01?1??0?213??01?1??0?????01002??1? 0?

0??

4? 已知向量组

A? a1?(0? 1? 1)T? a2?(1? 1? 0)T?

B? b1?(?1? 0? 1)T? b2?(1? 2? 1)T? b3?(3? 2? ?1)T?

??11301?r??11301?r??11301?(B, A)??02211?~?02211?~?02211?? ?11?110??02211??00000???????

5? 已知R(a1? a2? a3)?2? R(a2? a3? a4)?3? 证明

(1) a1能由a2? a3线性表示?

(2) a4不能由a1? a2? a3线性表示?

(2)假如a4能由a1? a2? a3线性表示? 则因为a1能由a2? a3线性表示? 故a4能由a2? a3线性表示? 从而a2? a3? a4线性相关? 矛盾? 因此a4不能由a1? a2? a3线性表示?

6? 判定下列向量组是线性相关还是线性无关?

(1) (?1? 3? 1)T? (2? 1? 0)T? (1? 4? 1)T?

(2) (2? 3? 0)T? (?1? 4? 0)T? (0? 0? 2)T?

??121?r??121?r??121? A??314?~?077?~?011?? ?101??022??000???????

(2)以所给向量为列向量的矩阵记为B? 因为

2?10 |B|?340?22?0? 002

7? 问a取什么值时下列向量组线性相关？

a1?(a? 1? 1)T? a2?(1? a? ?1)T? a3?(1? ?1? a)T? 解 以所给向量为列向量的矩阵记为A? 由

a1 |A|?1a??a(a?1)(a?1) 1?1a

?1(a1?b)??2(a2?b)?0?

????由此得 b??a1?a2??a1?(1?)a2? ?1??2?1??2?1??2?1??2

b?ca1?(1?c)a2? c?R?

9? 设a1? a2线性相关? b1? b2也线性相关? 问a1?b1? a2?b2是否一定线性相关？试举例说明之?

10? 举例说明下列各命题是错误的?

(1)若向量组a1? a2? ? ? ?? am是线性相关的? 则a1可由a2? ? ? ?? am线性表示?

(2)若有不全为0的数?1? ?2? ? ? ?? ?m使

?1a1? ? ? ? ??mam??1b1? ? ? ? ??mbm?0

?1a1? ? ? ? ??mam ??1b1? ? ? ? ??mbm ?0?

?1(a1?b1)? ? ? ? ??m(am?bm)?0?

(3)若只有当?1? ?2? ? ? ?? ?m全为0时? 等式

?1a1? ? ? ? ??mam??1b1? ? ? ? ??mbm?0

?1(a1?b1)??2(a2?b2)? ? ? ? ??m(am?bm)?0

(4)若a1? a2? ? ? ?? am线性相关, b1? b2? ? ? ?? bm亦线性相关? 则有不全为0的数? ?1? ?2? ? ? ?? ?m使

?1a1? ? ? ? ??mam?0? ?1b1? ? ? ? ??mbm?0

?1a1??2a2 ?0??1??2?2?

?1b1??2b2 ?0??1??(3/4)?2?

??1??2?0? 与题设矛盾?

11? 设b1?a1?a2? b2?a2?a3? b3?a3?a4? b4?a4?a1? 证明向量组b1? b2? b3? b4线性相关?

a1?b1?a2? a2?b2?a3? a3?b3?a4? a4?b4?a1?

?b1?b2?b3?a4

?b1?b2?b3?b4?a1?

12? 设b1?a1? b2?a1?a2? ? ? ?? br ?a1?a2? ? ? ? ?ar? 且向量组a1? a2? ? ? ? ? ar线性无关? 证明向量组b1? b2? ? ? ? ? br线性无关? 证明 已知的r个等式可以写成

?1?0(b1, b2, ? ? ? , br)?(a1, a2, ? ? ? , ar)?????0?11???0????????????1?1?? ????

1??

13? 求下列向量组的秩, 并求一个最大无关组?

(1)a1?(1? 2? ?1? 4)T? a2?(9? 100? 10? 4)T? a3?(?2? ?4? 2? ?8)T? 解 由

?19?2??19?2??19?2??2100?4?r?0820?r?010? (a1, a2, a3)??? ~~?1102??0190??000??44?8??0?320??000???????

(2)a1T?(1? 2? 1? 3)? a2T?(4? ?1? ?5? ?6)? a3T?(1? ?3? ?4? ?7)? 解 由

?1?2(a1, a2, a3)??1?3?41??141??1rr??1?3?~?0?9?5?~0?5?4??0?9?5??0?0?18?10??0?6?7?????41??9?5?? 00?00??

14? 利用初等行变换求下列矩阵的列向量组的一个最大无关组?

?25?75 (1)?75?25?

319494321753542043?132?? 134?48??

?25?75?75?25??1?0

(2)?

2?1?

3194943217535420

43?r2?3r1?25132?r3?3r1?0

?0134?r~4?r1

?048???1?

?1?? 3??1??

3117

121313

43??25r?r3?3?40~5?r3?r2?0

?05???

3117

120100

43?3?? 3?0??

1

201213025?14

?1?0?2?1?1201213025?14

1??1r?2r31

?1?~?03?r4?r1?0

?0?1???

1

2?20

21?1?2

25?52

1??1r?r32

?1?~?01?r3?r4?0

?0?2???

120021?20

25201??1?? ?2?0??

15? 设向量组

(a? 3? 1)T? (2? b? 3)T? (1? 2? 1)T? (2? 3? 1)T

3?r?1113??12a2?r?111

(a3, a4, a1, a2)??233b?~?01a?1?1?~?01a?1?1??

?1113??011b?6??002?ab?5???????而R(a1? a2? a3? a4)?2? 所以a?2? b?5?

16? 设a1? a2? ? ? ?? an是一组n维向量? 已知n维单位坐标向量e1? e2?? ? ?? en能由它们线性表示? 证明a1? a2? ? ? ?? an线性无关? 证法一 记A?(a1? a2? ? ? ?? an)? E?(e1? e2?? ? ?? en)? 由已知条件知? 存在矩阵K? 使

E?AK?

|E|?|A||K|?

R(e1? e2?? ? ?? en)?R(a1? a2? ? ? ?? an)?

17? 设a1? a2? ? ? ?? an是一组n维向量, 证明它们线性无关的充分必要条件是? 任一n维向量都可由它们线性表示? 证明 必要性? 设a为任一n维向量? 因为a1? a2? ? ? ?? an线性无关? 而a1? a2? ? ? ?? an? a是n?1个n维向量? 是线性相关的? 所以a能由a1? a2? ? ? ?? an线性表示? 且表示式是唯一的? 充分性? 已知任一n维向量都可由a1? a2? ? ? ?? an线性表示? 故单位坐标向量组e1? e2? ? ? ?? en能由a1? a2? ? ? ?? an线性表示? 于

n?R(e1? e2? ? ? ?? en)?R(a1? a2? ? ? ?? an)?n?

18? 设向量组a1? a2? ? ? ?? am线性相关? 且a1?0? 证明存在某个向量ak (2?k?m)? 使ak能由a1? a2? ? ? ?? ak?1线性表示? 证明 因为a1? a2? ? ? ?? am线性相关? 所以存在不全为零的数?1? ?2? ? ? ?? ?m? 使

?1a1??2a2? ? ? ? ??mam?0?

?k?0? ?k?1??k?2? ? ? ? ??m?0?

?1a1??2a2? ? ? ? ??kak?0?

ak??(1/?k)(?1a1??2a2? ? ? ? ??k?1ak?1)?

19? 设向量组B? b1? ? ? ?? br能由向量组A? a1? ? ? ?? as线性表示为

(b1? ? ? ?? br)?(a1? ? ? ?? as)K? 其中K为s?r矩阵? 且A组线性无关? 证明B组线性无关的充分必要条件是矩阵K的秩R(K)?r?

r?R(B)?R(AK)?min{R(A)? R(K)}?R(K)? 及 R(K)?min{r? s}?r?

Er? 充分性? 因为R(K)?r? 所以存在可逆矩阵C? 使KC???O???

(b1? ? ? ?? br)C?( a1? ? ? ?? as)KC?(a1? ? ? ?? ar)? 因为C可逆? 所以R(b1? ? ? ?? br)?R(a1? ? ? ?? ar)?r? 从而b1? ? ? ?? br线性无关?

20? 设

??1? ?2??3? ? ? ? ??n??2??1 ??3? ? ? ? ??n? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

????????? ? ? ? ???n123n?1

?0?1(?1, ?2, ? ? ? , ?n)?(?1, ?2, ? ? ? , ?n)?1?????1?101???1110???1???????????????1?1?1?? ????

0??

01|K|?1??1101???1110???1????????????????(?1)n?1(n?1)?0? ???0

21? 已知3阶矩阵A与3维列向量x满足A3x?3Ax?A2x? 且向量组x? Ax? A2x线性无关?

(1)记P?(x? Ax? A2x)? 求3阶矩阵B? 使AP?PB? 解 因为

AP?A(x? Ax? A2x)

?(Ax? A2x? A3x)

?(Ax? A2x? 3Ax?A2x)

?000? ?(x, Ax, Ax)?103?? ?01?1???2

?000?所以B??103?? ?01?1???

(2)求|A|?

22? 求下列齐次线性方程组的基础解系?

??x1?8x2?10x3?2x4?0 (1)?2x1?4x2?5x3?x4?0? ??3x1?8x2?6x3?2x4?0

0??1?8102?r?104 A??245?1? ~ ?01?3/4?1/4?? ?386?2??000?0????

x1??4x3 ??x?(3/4)x?(1/4)x? ?234

?1?(?16? 3? 4? 0)T? ?2?(0? 1? 0? 4)T?

??2x1?3x2?2x3?x4?0 (2)?3x1?5x2?4x3?2x4?0? ??8x1?7x2?6x3?3x4?0

?2?3?21?r?102/19?1/19? A??354?2? ~ ?0114/19?7/19?? ?876?3??000?0????

x1??(2/19)x3?(1/19)x4 ??x??(14/19)x?(7/19)x? ?234

?1?(?2? 14? 19? 0)T? ?2?(1? 7? 0? 19)T?

(3)nx1 ?(n?1)x2? ? ? ? ?2xn?1?xn?0.

xn??nx1?(n?1)x2? ? ? ? ?2xn?1?

?1?(1? 0? 0? ? ? ?? 0? ?n)T?

?2?(0? 1? 0? ? ? ?? 0? ?n?1)T?

? ? ??

?n?1?(0? 0? 0? ? ? ?? 1? ?2)T?

2?213, 求一个4?2矩阵B, 使AB?0, 且 23? 设A???9?528????

R(B)?2.

2?213 ~ ?10?1/81/8?? A???9?528???01?5/8?11/8?????r

x1?(1/8)x3?(1/8)x4 ??x?(5/8)x?(11/8)x? ?234

?1?(1? 5? 8? 0)T? ?2?(?1? 11? 0? 8)T?

?1?5 因此所求矩阵为B??8?0??1?11?? 0?8??

24? 求一个齐次线性方程组, 使它的基础解系为

?1?(0? 1? 2? 3)T ? ?2?(3? 2? 1? 0)T ?

?x1??0??x1?3k23???x??1??2??x2?k1?2k22, 即?k?k?x?1?2?2?1??x?2k?k? (k1? k2?R)? ?3??3??0??x3?3k12???41?x4???

?2x1?3x2?x4?0? ?x?3x?2x?0?134

25? 设四元齐次线性方程组

x1?x2?0 I? ??x?x?0 ? II? ?24?x1?x2?x3?0? ?x?x?x?0?234

x??x 解 (1)由方程I得??x1?x4? ?24

?1?(0? 0? 1? 0)T? ?2?(?1? 1? 0? 1)T?

x1??x4 由方程II得??x?x?x? ?234

?1?(0? 1? 1? 0)T? ?2?(?1? ?1? 0? 1)T?

(2) I与II的公共解就是方程

?x1?x2?0?x?x?0 III? ?24 x1?x2?x3?0?x?x?x?0?234

?1?0 A??1?0?11?11001?10??1r??1? ~0 0??0?1???0010001?0?1?? 1?2?

00??

??x1??x4 ?x2?x4? ??x3?2x4

26? 设n阶矩阵A满足A2?A? E为n阶单位矩阵, 证明

R(A)?R(A?E)?n?

R(A)?R(A?E)?R(A)?R(E?A)?R(A?E?A)?R(E)?n?

27? 设A为n阶矩阵(n?2)? A*为A的伴随阵? 证明

n 当R(A)?n??R(A*)??1 当R(A)?n?1? ??0 当R(A)?n?2

|AA*|?||A|E|?|A|?0? |A*|?0?

AA*?|A|E?0?

28? 求下列非齐次方程组的一个解及对应的齐次线性方程组的基础解系?

??x1?x2?5 (1)?2x1?x2?x3?2x4?1? ??5x1?3x2?2x3?2x4?3

?11005?r?1010?8?B??21121? ~ ?01?1013?? ?53223??00012?????

??x1??x3?8

?x2? x3?13? ??x4? 2

??x1??x3

?x2? x3? ??x4?0

??x1?5x2?2x3?3x4?11 (2)?5x1?3x2?6x3?x4??1? ??2x1?4x2?2x3?x4??6

?1?52?311?r?109/7?1/21? B??536?1?1? ~ ?01?1/71/2?2?? ?2421?6??00000?????

?x1??(9/7)x3?(1/2)x4?1? ?x?(1/7)x?(1/2)x?2?234

??(1? ?2? 0? 0)T?

?x1??(9/7)x3?(1/2)x4? ?x?(1/7)x?(1/2)x?234

?1?(?9? 1? 7? 0)T? ?2?(1? ?1? 0? 2)T?

29? 设四元非齐次线性方程组的系数矩阵的秩为3? 已知?1? ?2? ?3是它的三个解向量? 且

?1?(2? 3? 4? 5)T? ?2??3?(1? 2? 3? 4)T?

2?1?(?2??3)?(?1??2)?(?1??3)? (3? 4? 5? 6)T

x?k(3? 4? 5? 6)T?(2? 3? 4? 5)T? (k?R)?

30? 设有向量组A? a1?(?? 2? 10)T? a2?(?2? 1? 5)T? a3?(?1? 1?

4)T? 及b?(1? ?? ?1)T? 问?? ?为何值时

(1)向量b不能由向量组A线性表示?

(2)向量b能由向量组A线性表示? 且表示式唯一?

(3)向量b能由向量组A线性表示? 且表示式不唯一? 并求一般表示式?

1???1?2?1?r??1?2? 解 (a3, a2, a1, b)??112??~ ?0?11????1?? ?4510?1??004???3??????

(1)当???4? ??0时? R(A)?R(A? b)? 此时向量b不能由向量组A线性表示?

(2)当???4时? R(A)?R(A? b)?3? 此时向量组a1? a2? a3线性无关? 而向量组a1? a2? a3? b线性相关? 故向量b能由向量组A线性表示? 且表示式唯一?

(3)当???4? ??0时? R(A)?R(A? b)?2? 此时向量b能由向量组A线性表示? 且表示式不唯一?

??1?2?41?r?10?21?(a3, a2, a1, b)??1120?~ ?013?1?? ?4510?1??0000?????

?x1??2??1??2c?1? ?x2??c??3????1????3c?1?? c?R? ?x??1??0??c???3??????

31? 设a?(a1? a2? a3)T? b?(b1? b2? b3)T? c?(c1? c2? c3)T? 证明三直线

l1? a1x?b1y?c1?0?

l2? a2x?b2y?c2?0? (ai2?bi2?0? i?1? 2? 3) l3? a3x?b3y?c3?0?

???a1x?b1y?c1?0?a1x?b1y??c1

?a2x?b2y?c2?0? 即?a2x?b2y??c2 ???a3x?b3y?c3?0?a3x?b3y??c3

32? 设矩阵A?(a1? a2? a3? a4)? 其中a2? a3? a4线性无关? a1?2a2? a3? 向量b?a1?a2?a3?a4? 求方程Ax?b的通解? 解 由b?a1?a2?a3?a4知??(1? 1? 1? 1)T是方程Ax?b的一个解?

x?c(1? ?2? 1? 0)T?(1? 1? 1? 1)T? c?R?

33? 设?*是非齐次线性方程组Ax?b的一个解, ?1? ?2? ? ? ?? ?n?r ?是对应的齐次线性方程组的一个基础解系, 证明?

(1)?*? ?1? ?2? ? ? ?? ?n?r线性无关?

(2)?*? ?*??1? ?*??2? ? ? ?? ?*??n?r线性无关?

(2)显然向量组?*? ?*??1? ?*??2? ? ? ?? ?*??n?r与向量组?*?

?1? ?2? ? ? ?? ?n?r可以相互表示? 故这两个向量组等价? 而由(1)知向量组?*? ?1? ?2? ? ? ?? ?n?r线性无关? 所以向量组?*? ?*??1? ?*??2? ? ? ?? ?*??n?r也线性无关?

34? 设?1? ?2? ? ? ?? ?s是非齐次线性方程组Ax?b的s个解? k1? k2? ? ? ?? ks为实数? 满足k1?k2? ? ? ? ?ks?1. 证明

x?k1?1?k2?2? ? ? ? ?ks?s

35? 设非齐次线性方程组Ax?b的系数矩阵的秩为r? ?1? ?2? ? ? ?? ?n?r?1是它的n?r?1个线性无关的解? 试证它的任一解可表示为

x?k1?1?k2?2? ? ? ? ?kn?r?1?n?r?1? (其中k1?k2? ? ? ? ?kn?r?1?1). 证明 因为?1? ?2? ? ? ?? ?n?r?1均为Ax?b的解? 所以?1??2??1? ?2??3??1? ? ? ?? ?n?r?? n?r?1??1均为Ax?b的解? 用反证法证? ?1? ?2? ? ? ?? ?n?r线性无关?

?1?1? ?2?2? ? ? ? ? ? n?r ? n?r?0?

?(?1??2? ? ? ? ??n?r)??1??2? ? ? ? ??n?r?0? 矛盾? 因此?1? ?2? ? ? ?? ?n?r线性无关? ?1? ?2? ? ? ?? ?n?r为Ax?b的一个基础解系?

x??1?k2?1?k3?2? ? ? ? ?kn?r?1?n?r

?k2(?2??1)?k3(?3??1)? ? ? ? ?kn?r?1(?n?r?1??1)? x??1(1?k2?k3 ? ? ? ?kn?r?1)?k2?2?k3?3? ? ? ? ?k n?r?1?n?r?1? 令k1?1?k2?k3 ? ? ? ?kn?r?1? 则k1?k2?k3 ? ? ? ?kn?r?1?1? 于是 x?k1?1?k2?2? ? ? ? ?kn?r?1?n?r?1?

36? 设

V1?{x?(x1? x2? ?????? xn)T | x1? ?????? xn?R满足x1?x2? ???????xn?0}? V2?{x?(x1? x2? ?????? xn)T | x1? ?????? xn?R满足x1?x2? ???????xn?1}? 问V1? V2是不是向量空间？为什么？

??(a1? a2? ?????? an)T ?V1? ??(b1? b2? ?????? bn)T ?V1? ???R?

37? 试证? 由a1?(0? 1? 1)T? a2?(1? 0? 1)T? a3?(1? 1? 0)T所生成的向量空间就是R3.

11|A|?01??2?0? 10

38? 由a1?(1? 1? 0? 0)T? a2?(1? 0? 1? 1)T所生成的向量空间记作V1,由b1?(2? ?1? 3? 3)T? b2?(0? 1? ?1? ?1)T所生成的向量空间记作V2, 试证V1?V2.

?1?1 (A, B)??0?0?10112?1330??1r?1? ~0 ?1??0?0?1???1?1002?3000?1?? 0?

0??

39? 验证a1?(1? ?1? 0)T? a2?(2? 1? 3)T? a3?(3? 1? 2)T为R3的一个基, 并把v1?(5? 0? 7)T? v2?(?9? ?8? ?13)T用这个基线性表示. 解 设A?(a1? a2? a3)? 由

123|(a1, a2, a3)|??111??6?0? 032

??x1?2x2?3x3?5

??x1?x2?x3?0? ??3x2?2x3?7

??x1?2x2?3x3??9

??x1?x2?x3??8? ??3x2?2x3??13

40? 已知R3的两个基为

a1?(1? 1? 1)T? a2?(1? 0? ?1)T? a3?(1? 0? 1)T? b1?(1? 2? 1)T? b2?(2? 3? 4)T? b3?(3? 4? 3)T? 求由基a1? a2? a3到基b1? b2? b3的过渡矩阵P? 解 设e1? e2? e3是三维单位坐标向量组? 则

?111? (a1, a2, a3)?(e1, e2, e3)?100?? ?1?11???

?1 (e1, e2, e3)?(a1, a2, a3)?1?1?

?1于是 (b1, b2, b3)?(e1, e2, e3)?2?1?10?11?0?? 1??23?34? 43??

?1?1?111??123? ?(a1, a2, a3)?100??234?? ?1?11??143?????

?111??123??234? P??100??234???0?10?? ?1?11??143???10?1???????

?1

1? 验证所给矩阵集合对于矩阵的加法和乘数运算构成线性空间? 并写出各个空间的一个基?

(1) 2阶矩阵的全体S1?

(A?B)?S1? kA?S1?

1?1???0

(2)主对角线上的元素之和等于0的2阶矩阵的全体S2?

?ab? B???de?? A? B?S? 因为 解 设A???ca???fd?2????

?(a?d)c?b? A?B???c?aa?d??S2? ??

?kakb?S? kA???kcka??2??

10? ???0?1???0?1??2?0??

(3) 2阶对称矩阵的全体S3. 1?? ???0?3?10???0? 0??

(A?B)T?AT?BT?A?B? (A?B)?S3?

(kA)T?kAT?kA? kA?S3?

1?1???0

2? 验证? 与向量(0? 0? 1)T不平行的全体3维数组向量? 对于数组向量的加法和乘数运算不构成线性空间?

3? 设U是线性空间V的一个子空间? 试证? 若U与V的维数相等? 则U?V?

4? 设Vr是n维线性空间Vn的一个子空间? a1? a2? ???? ar是Vr的一个基? 试证? Vn中存在元素ar?1? ???? an? 使a1? a2? ???? ar, ar?1? ???? an成为Vn的一个基?

5? 在R3中求向量??(3? 7? 1)T在基?1?(1? 3? 5)T? ?2?(6? 3? 2)T? ?3?(3? 1? 0)T下的坐标?

(?1? ?2? ?3)?(?1? ?2? ?3)A?

(?1? ?2? ?3)?(?1? ?2? ?3)A?1?

?163???26?3??1其中A??331?? A??5?158?? ?520???928?15?????

?3??3?因为 ??(?1, ?2, ?3)?7??(?1, ?2, ?3)A?1?7? ?1??1?????

??26?3??3? ?(?1, ?2, ?3)?5?158??7? ??928?15??1?????

?33? ?(?1, ?2, ?3)??82?? ?154???

6? 在R3取两个基 ?1?(1? 2? 1)T? ?2?(2? 3? 3)T? ?3?(3? 7? 1)T? ?1?(3? 1? 4)T? ?2?(5? 2? 1)T? ?3?(1? 1? ?6)T? 试求坐标变换公式? 解 设?1? ?2? ?3是R3的自然基? 则 (?1? ?2? ?1)?(?1? ?2? ?3)B? (?1? ?2? ?3)?(?1? ?2? ?1)B?1? (?1? ?2? ?1)?(?1? ?2? ?3)A?(?1? ?2? ?1)B?1A?

?121??351?其中 A??237?? B??121?? ?131??41?6?????

?x1??x1???(?1, ?2, ?3)?x2??(?1, ?2, ?3)B?1A?x2?? ?x??x??3??3?

?1319181???4?xx?1??1???x1?????B?1A?x2????9?13?63??x2?? ?x2?x???x?2??x???3??3??3?99?710?4??

7? 在R4中取两个基

e1?(1?0?0?0)T? e2?(0?1?0?0)T? e3?(0?0?1?0)T? e4?(0?0?0?1)T? ?1?(2?1??1?1)T? ?2?(0?3?1?0)T? ?3?(5?3?2?1)T? ?3?(6?6?1?3)T? (1)求由前一个基到后一个基的过渡矩阵? 解 由题意知

?2?1

(?1, ?2, ?3, ?4)?(e1, e2, e3, e4)?

?1?1?

031053216?6?? 1?3??

?2

?1A??

?1?1?

031053216?6?? 1?3??

(2)求向量(x1? x2? x3? x4)T在后一个基下的坐标? 解 因为

?x1??x1?

?x???

?1x22

??(e1, e2, e3, e4)???(?1, ?2, ?3, ?4)A???

?x3??x3??x4??x4?

?y1??2

?y??0?y2???5?3??6?y4??

1336?1121

1?0?1?3??

?1

?x1??12

?x?1?1?x2???9?3?27??7

??x4?

9

120?3?27?909

?33??x1??23??x2?? ?18??x3?

??26???x4?

(3)求在两个基下有相同坐标的向量. 解 令

?121?1?27?9

??79120?3?27?909?33??x1??x1??23??x2???x2?? ?18??x3??x3?????26???x4??x4?

?x1??1??x??1?解方程组得?2??k??(k为常数)? x1?3????x4??1?

x??x?的几何意义? 其中 8? 说明xOy平面上变换T??y??A?y?????

?1 (1)A???0?

x???1T??y???0???

0 (2)A???0?0?? 1??0??x????x?? ?y??y?1??????所以在此变换下T(?)与?关于y轴对称?

x??0T??y???0???

0 (3)A???1?1?? 0??0??x???0?? ????1???y??y?所以在此变换下T(?)是?在y轴上的投影?

x??0T??y???1???

0 (4)A????1?

x??0T??y????1???1??x???y?? ????0???y???x?

9? n阶对称矩阵的全体V对于矩阵的线性运算构成一个n(n?1)维线性空间. 给出n阶矩阵P? 以A表示V中的任一元素? 2

T(A?B)?PT(A?B)P?PT(A?B)TP

?[(A?B)P]TP?(AP?BP)TP

?(PTA?PTB)P?PTAP?PTBP?T(A)?T(B)?

T(kA)?PT(kA)P?kPTAP?kT(A)?

10? 函数集合

V3?{??(a2x2?a1x?a0)ex | a2? a1? a0 ?R}

?1?x2ex? ?2?xex? ?3?ex?

?1?D(?1)?2xex?x2ex?2?2??1? ?2?D(?2)?ex?xex??3??2? ?3?D(?3)?ex??3?

?100?由 (?1, ?2, ?3)?(?1, ?2, ?3)?210?? ?011???

?100?知即D在基?1? ?2? ?3下的矩阵为P??210?? ?011???

11? 2阶对称矩阵的全体

x1x2?V3?{A???xx?|x1,x2,x3?R} ?23?

1A1???0?0?? A??0

2?10???

1T(A)???1?1?? A??03?00???0?. 1??在V3中定义合同变换 0?A?11?, ?01?1????

1? T(A)??11?0??1?1???00??11???11??A?A?A? ????0???01??11?1231 T(A2)???1?

1 T(A3)???1?0??0?11???0??0?01???1??1?01???1???0?11???1??A?2A? 32??20??A? 1??30??11???0?01??01?????

?100?故 (T(A1), T(A2), T(A3))?(A1, A2, A3)?110?? ?121???

?100?从而? T在基A1? A2? A3下的矩阵A??110?. ?121???