线性代数-北京邮电大学出版社-戴赋祥

?1?1A???3??1?1?0??0??0?11?13?12005?28?95?700?1??1?15?1??02?74?3?r3?r2r?r21???????????r4?2r23?3r1??1?r02?74r4?r1???7??04?148?

?1?4??r(A)?2.0??0?∴ 其基础解系含有4?R(A)?2个解向量.

3?x1???x3?x??2?2?x1?x2?5x3?x4?0???7?????x3?2???2x2?7x3?4x4?0?x3??x3???x?4???基础解系为

??3??x4???2???1??????2?7????2x4?x3?x4?? ??2??0???1???????1??x4??0????3???2????7?,?2??1?????0??

(3)

??1???2???. ?0????1??1122A??2345???3568?112r3?2r2?????010???000得同解方程组

7??11227?r2?2r1?0101?14?????0??r3?3r1???0???0202?21??

27?1?14??07???x1?x2?2x3?2x4?7x5?0,?x2?x4?14x5?0, ??7x5?0?x5?0.?取??x3??1??0????0?,?1?得基础解系为 x?4?????(?2，0，1，0，0)T,(?1，?1，0，1，0).

(4) 方程的系数矩阵为

?12?2A??12?1???24?7?12r3?3r2?????00???002?1??12?22?1?r2?r1?0011?1?????3?2??r3?2r1???11???00?3?33??

?22?1?R(A)?2,11?1??000???x2??0??1??0??x???0?,?0?,?1?, ?4?????????1????0????0???x5???∴ 基础解系所含解向量为n?R(A)=5?2=3个

?x2???取x4为自由未知量 ????x5???3???2???4??0??1??0???????得基础解系 ?1?,?0?,??1?.

???????0??0??1???1????0????0??3. 解下列非齐次线性方程组.

?x1?x2?2x3?1,?2x1?x2?x3?x4?1,?2x?x?2x?4,??123(1) ? (2) ?4x1?2x2?2x3?x4?2,

?2x?x?x?x?1;?x1?2x2?3,1234???4x1?x2?4x3?2;?x1?x2?x3?x4?x5?7,x?2x?x?x?1,?1234?3x?2x?x?x?3x??2,??12345(3) ?x1?2x2?x3?x4??1, (4) ?

?x?2x?x?x?5;?x2?2x3?2x4?6x5?23,234?1??5x1?4x2?3x3?3x4?x5?12.【解】

（1） 方程组的增广矩阵为

?11?2?1(A?b)???1?2??41?11?0?3??00??0021??11?0?324?r?2r21??????r3?r103?r4?4r1?0?3??42??0?321??1?01?r4?r3?22?2????????000????2?4??02?2?2?41?3001?2?r3?r2?????r4?r22???2?

21??22??12??00?得同解方程组

x3?2,??x1?x2?2x3?1?2?2x3???3x?2x?2?x???2, ??223?3??x?23???x1?1?x2?2x3??1.(2) 方程组的增广矩阵为

?21?111??21?111?r3?r1?000?10?

(A?b)??42?212???????r2?2r1?????21?1?11???000?20??得同解方程组

?2x1?x2?x3?x4?1,??x4?0 ?x4?0,???2x4?0,?即

?2x1?x2?x3?1, ?x?0.?4令x1?x3?0得非齐次线性方程组的特解

xT=(0，1，0，0)T.

又分别取

?x2??1??0??x???0?,?1? ?3?????得其导出组的基础解系为

TT1??1???,1,0,0?;??2?∴ 方程组的解为

1??2???,0,1,0?,

?2??1??1???0??2??2??1?????x????k1?1??k2?0?.?0??0??1????????0?0??????0??k1,k2?R

1??1?2111??1?211??r2?r1??000?2?2?

(3) 1?21?1?1?????r3?r1????4??1?2115???0000?R(A)?R(A)∴ 方程组无解.

(4) 方程组的增广矩阵为

?1?3(A?b)???0??5?1?0r3?r2?????r4?r2?0??0分别令

11112210010016100211?3433?1?1?2?27?7??11111?0?1?2?2?6?23??2?r3?3r1???????r4?5r1?0122623?23????12?0?1?2?2?6?23??

17??6?23??,00??00??x3??0??1??0??x???0?,?0?,?1? ?4?????????1????0????0???x5???得其导出组??x1?x2?x3?x4?x5?0的解为

?x?2x?2x?6x?0345?2?5??1??1???6???2???2???????k1?0??k2?1??k3?0???????00?????1?????1???0???0??令x3?x4?x5?0,

得非齐次线性方程组的特解为：xT=(?16,23,0,0,0)T,

∴ 方程组的解为

k1,k2,k3?R.

??16??5??1??1??23???6???2???2?????????x??0??k1?0??k2?1??k3?0?

????????000???????1??????0???1???0???0??其中k1,k2,k3为任意常数.

4. 某工厂有三个车间，各车间相互提供产品（或劳务），今年各车间出厂产量及对其它车间

的消耗如下表所示. 车间 消耗系数 车间 1 2 0.1 0.2 0.2 0.2 0.45 0.3 1 2 3 出厂产量 （万元） 22 0 总产量 （万元） x1 x2