(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
1 2 3 出厂产量 （万元） 22 总产量 （万元） x1 0.1 0.2 0.45
2 3 0.2 0.5 0.2 0 0.3 0.12 0 55.6 x2 x3 表中第一列消耗系数0.1,0.2,0.5表示第一车间生产1万元的产品需分别消耗第一，二，三车间0.1万元，0.2万元，0.5万元的产品；第二列，第三列类同，求今年各车间的总产量.
解：根据表中数据列方程组有
?x1? 0.1x1?0.2x2? 0.45x3?22,??x2? 0.2x1?0.2x2?0.3x3?0, ?x3?0.5x1?0.12x3?55.6,??0.9x1?0.2x2? 0.45x3?22,?即 ? 0.2x1?0.8x2?0.3x3?0,
?0.5x?0.88x??55.6,13??x1?100,?解之 ?x2?70,
?x?120;?35. ?取何值时，方程组
??x1?x2?x3?1,??x1??x2?x3??, ?x?x??x??2,3?12(1)有惟一解，(2)无解，(3)有无穷多解，并求解.
【解】方程组的系数矩阵和增广矩阵为
??A??1???11?11?;1???????B??1???12111?1?1?,??? ?2??|A|=(??1)(??2).
(1) 当?≠1且?≠?2时，|A|≠0，R(A)=R(B)=3.
∴ 方程组有惟一解
???11(??1)2x1?,x2?,x3?.
??2??2(??2)（2） 当?=?2时，
??2B??1???1?1?0???01??1?21?2?r3?r1r2?r1??????2111??????21?2?r???2?2r1?1?24???11?24??
?21?2??1?21?2???0?33?3?,?33?3????3?36????0003??11
R(A)≠R(B),∴ 方程组无解. (3) 当?=1时
?1111??1111?r2?r1?0000?
B??1111?????r?r??31?????1111???0000??R(A)=R(B)<3,方程组有无穷解.
得同解方程组
?x1??x2?x3?1,?x2?x2, ??x3?x3.?∴ 得通解为
?x1???1???1??1??x??k?1??k?0???0?, k,k?R.
12?2?1??2????????0???1????0???x3??6. 齐次方程组
??x?y?z?0,??x??y?z?0, ?2x?y?z?0?当?取何值时，才可能有非零解?并求解. 【解】方程组的系数矩阵为
??A??1???21???1?? ?11??1|A|=(??4)(??1)
当|A|=0即?=4或?=?1时，方程组有非零解.
(i) 当?=4时,
?411??14r2?r1A??14?1??????41??????2?11???2?1?141r2??0?35???1?r3?3??0?3得同解方程组
?1??1r2?4r1?0????1?r?2r?31??1???0?1??1r3?r2?????01????1???0?1??155???93??
4?1??31??00??4
?1???3??x1????x1?4x2?x3?0????x?k?1?.k?R ??3x?x?0??2?23???x??3??3????1?(ii) 当?=?1时,
??111??1?1?1??1?1?1?r2?r1r2?r1?000?
A??1?1?1???????111?????????r3?2r1??????2?11???2?11???013??得
?x1??2x3,?x1?x2?x3?0???x2??3x3, ??x2?3x3?0?x?x3?3∴ (x1,x2,x3)T=k·(?2,?3,1)T.k∈R
7. 当a,b取何值时，下列线性方程组无解，有惟一解或无穷多解？在有解时，求出其解.
?x1?2x2?3x3?x4?1?x1?x2?x3?x4?0?x?x?2x?3x?1?x?2x?2x?1?12?34234（1） ? (2) ?
?3x1?x2?x3?2x4?a??x2?(a?3)x3?2x4?b???2x1?3x2?x3?bx4??6?3x1?2x2?x3?ax4??1【解】方程组的增广矩阵为
(1)
?12?11(A?b)???3?1??23?12?0?1??00??0032?1?13?1?3?61??1?0r2?r131?r?3r31??????r4?2r1?0?2a???b?6??0?11??1?040???????0?27a?3???b?2?8??0?12?1?12003?1?73?30?7?10?1?11?40?r3?7r2?????1a?3?r4?r2?b?2?8?
?11?40??.?27a?3??b?52?2a?2??1(i) 当b≠?52时,方程组有惟一解
a4(a?1)a?326(a?1)?,x2??,3b?523b?52
a?318(a?1)2(a?1)x3???,x4??.3b?52b?52x1?(ii) 当b=?52,a≠?1时，方程组无解.
(iii) 当b=?52，a=?1时，方程组有无穷解. 得同解方程组
?x1?2x2?3x3?x4?1???x2?x3?4x4?0 (*) ??3x?27x??434??x1?2x2?3x3?x4?0?其导出组??x2?x3?4x4?0的解为
??3x?27x?034??x1?2x4,?x?13x?24??x3??9x4,??x4?x4.非齐次线性方程组（*）的特解为
?x1??2??x??13??2??k??.k?R ?x3???9?????x?1??4??5??3??x1????x??35?2取x4=1, ????3?.
?x3???23??????x4??3??1???∴ 原方程组的解为
?5??3??2????13??35?x??3??k??.??9???23??????1??3??1??? (2)
k?R