线性代数习题解答(同济大学（第四版）)(4)

第二章 矩阵及其运算

1．已知线性变换：

?x1?2y1?2y2?y3,??x2?3y1?y2?5y3, ?x?3y?2y?3y,123?3求从变量x1,x2,x3到变量y1,y2,y3的线性变换．

解

?x1??221??y1???????由已知：?x2???315??y2?

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

?y??323??x??3?y?2?4??2????3????3??y1??7x1?4x2?9x3??y2?6x1?3x2?7x3 ?y?3x?2x?4x123?3

2．已知两个线性变换

?x1?2y1?y3,?y1??3z1?z2,??x??2y?3y?2y, ?2?y2?2z1?z3, 123?x?4y?y?5y,?y??z?3z,12323?3?3求从z1,z2,z3到x1,x2,x3的线性变换．

解 由已知

?x1??201??y1??20????????x2????232??y2????23?x??415??y??41??2???3????613??z1???????12?49??z2? ??10?116??z????3??x1??6z1?z2?3z3?所以有 ?x2?12z1?4z2?9z3

?x??10z?z?16z123?3?111??1???1?1?, B???13．设A??1?1?11??0???1???310??z1??????2??201??z2? ????5???0?13??z3?23???24?, 51??16

求3AB?2A及A解

TB.

23??11?111??1?????3AB?2A?3?11?1???1?24??2?11?1?11??0?51??????1?1?058??111???213??????3?0?56??2?11?1????2?17?290??1?11??429?????23??058??111??1??????TAB??11?1???1?24???0?56?

?1?11??0??51??????290?

4．计算下列乘积：

1???1? 1??22??20? ?2???431??7??3??2?????????(1)?1?23??2?; (2)?1,2,3??2?; (3)?1???1,2?; ?570??1??1??3?????????1??13???2140??0?12?(4)??1?134???1?31?; ????40?2?????a11a12a13??x1?????(5)(x1,x2,x3)?a12a22a23??x2?;

?aa??x?a?132333??3?1??1210??103?????0101??012?1?(6)

?0021??00?23?. ??0003????000?3??????解

?431??7??4?7?3?2?1?1??35?????????(1)?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?2?2?3?1)?(10)

?1???17

?2?(?1)2?2???24??2???????(3)?1???12???1?(?1)1?2????12? ?3??3?(?1)3?2???36???????1??13???2140??0?12??6?78?(4)??1?134???1?31????20?5?6?? ???????40?2????a11a12a13??x1?????(5)?x1x2x3??a12a22a23??x2?

?a??x?aa?132333??3???a11x1?a12x2?a13x3a12x1?a22x2?a23x3a13x1?a23x2?a33x3? ?x1???222?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3 ??x2??a11x1?x??3?1??1252??1210??103???????0101??012?1??012?4?(6)

?0021??00?23???00?43? ??0003????000?3????000?9????????

?12??10?5．设A???13??, B???12??，问：

????(1)AB?BA吗?

222(2)(A?B)?A?2AB?B吗?

22(3)(A?B)(A?B)?A?B吗?

解

?12??10???, B??????13??12??34??12?则AB???46?? BA???38?? ?AB?BA

?????22??22??814?2(2) (A?B)???25????25?????1429??

???????38??68??10??1016?22但A?2AB?B???411?????812?????34?????1527??

????????222故(A?B)?A?2AB?B

?22??02??06?(3) (A?B)(A?B)???25????01?????09??

??????(1)A???18

?38??10??28?A2?B2???411?????34?????17??

??????22故 (A?B)(A?B)?A?B

而

6．举反列说明下列命题是错误的:

?0,则A?0; 2（２）若A?A,则A?0或A?E; （３）若AX?AY,且A?0,则X?Y.

?01?2?解 (1) 取A?? A?0,但A?0 ?00????11?2A?A,但A?0且A?E ?(2) 取A?? ?00????10??11??11??????(3) 取A?? X?? Y?? ?????00???11??01?AX?AY且A?0 但X?Y

（１）若A

2?10?23k?,求. A,A,?,A???1??10??10??10?2解 A????1?????1?????2?1??

???????10??10??10?32 A?AA???2?1?????1?????3?1??

???????10?k利用数学归纳法证明: A???k?1??

??当k?1时,显然成立,假设k时成立,则k?1时

0??10??10??1kkA?AA???k?1?????1?????(k?1)?1??

???????10?k由数学归纳法原理知:A???k?1??

??7．设A???

??10???k8．设A??0?1?,求A.

?00????解 首先观察

2?2?1???10???10????????22A??0?1??0?1???0?2??

?00???00???00?2???????

19

??33?23????3232?3?? A?A?A??0?03?0???k(k?1)k?2??kk?1????k?2??kkk?1由此推测 A??0?k?? (k?2)

k?0?0?????用数学归纳法证明:

当k?2时,显然成立.

假设k时成立，则k?1时，

k(k?1)k?2??kk?1?k?????10??2????k?1kkk?1A?A?A??0?k???0?1?

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

k?1?0?0?????k(k?1)k?2??kk?1?k????2??kkk?1由数学归纳法原理知: A??0?k??

k?0?0?????

9．设A,B为n阶矩阵，且A为对称矩阵，证明B证明 已知：ATTAB也是对称矩阵.

?A

TTTTTT则 (BTAB)?B(BTA)?BAB?BAB

T从而 BAB也是对称矩阵.

AB?BA.

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

?A BT?B

TTT充分性：AB?BA?AB?BA?AB?(AB) 即AB是对称矩阵.

TTT必要性：(AB)?AB?BA?AB?BA?AB.

证明 由已知：A

11．求下列矩阵的逆矩阵：

T20

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