解 对系数矩阵A进行初等行变换 有 13?17?20??17?00?? A?34?5?2?33?411?13?7?21??17???2?~?016???03????00?3171?19170000

?x?3x?13x?1173174?1920于是 ?x2?x3?x41717?x?x?x3?x3?44故方程组的解为

?3???13??17??x1??17??x??19??20?2 ???k1???k2???(k1

17??x3??17??0?x4??1????0??1? k2为任意常数)

13

求解下列非齐次线性方程组:

有

??4x1?2x2?x3?2 (1)?3x1?1x2?2x3?10??11x1?3x2?8 解 对增广矩阵B进行初等行变换

B于是R(A)

?42?12??13?3?8??3?1210?~0?101134?11308??000?6?????2 而R(B)

3 故方程组无解

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

解 对增广矩阵B进行初等行变换 有

B?2314??1?1?24?5??0?38?213?~?0?4?19?6??0???

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

有

解 对增广矩阵B进行初等行变换 B?21?111??11/2?1/201/2??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??2??y??即 ???k1?1??k2?0???0?(k1 k2为任意常数)

z0??1??0??w????0??0????0????????2x?y?z?w?1 (4)?3x?2y?z?3w?4??x?4y?3z?5w??2

有

解 对增广矩阵B进行初等行变换 B?21?111??10?1/7?1/76/7??3?21?34?~01?5/79/7?5/7??14?35?2??00000?????x?1z?1w?6?777?595于是 ?y?z?w?777?z?z?w?w?

?1??1??6??7??7??x??7??9??5??y??5?即 ???k1???k2???????(k1

z7??7????w??700???1?????010?????? k2为任意常数)

14 写出一个以

?2???2???3??4?x?c1???c2??

10?0??1?????为通解的齐次线性方程组 解 根据已知

可得

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

与此等价地可以写成 ?x1?2c1?c2?x??3c1?4c2 ?2x?c?x3?c1?42

x1?2x3?x4或 ??x??3x?4x?234x1?2x3?x4?0或 ??x?3x?4x?0?234

这就是一个满足题目要求的齐次线性方程组 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