内容发布更新时间 : 2024/5/24 2:13:51星期一 下面是文章的全部内容请认真阅读。

??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? 有

?42?12??13?3?8? B??3?1210?~?0?101134??

?11308?000?6????于是R(A)?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

?2314??1?1?24?5??0 B??~

38?213??0?4?19?6??0?????x??2z?1于是 ?y?z?2?

??z?z01002?100?1?2?? 0?0???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 解 对增广矩阵B进行初等行变换? 有

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

?21?1?11??00?000?????x??1y?1z?1?222?于是 ?y?y?

?z?z??w?01??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 解 对增广矩阵B进行初等行变换? 有

?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?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? k2为任意常数)?

z7??7????w??700???1??????0??1??0? 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 ?2?

x3?c1?x?c?42x1?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??x1?x2??x3?? (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时方程组有唯一解.

(2)要使方程组无解? 必须R(A)?R(B)? 故 (1??)(2??)?0? (1??)(??1)2?0? 因此???2时? 方程组无解?

(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? 因此当??1时? 方程组有无穷多个解.

16? 非齐次线性方程组

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

2??x1?x2?2x3??当?取何值时有解？并求出它的解?

1?21????211?2????2?? 解 B?1?21?~?01?1?(??1)??

?11?2?2?3???000(??1)(??2)???要使方程组有解? 必须(1??)(??2)?0? 即??1? ???2?

当??1时?

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

?11?21??0000?????方程组解为

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????? 当???2时?

??211?2??10?12? B??1?21?2?~?01?12??

?11?24??0000?????方程组解为

x1?x3?2??x?x?2 ??x1?x3?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问?为何值时? 此方程组有唯一解、无解或有无穷多解? 并在有