?x1?1?k?x?k2?? ?x3?0?x?k?4??x5??2?2k其中k为任意常数.
2）对方程组德增广矩阵作行初等变换，有
0?321??120?321??12?1?1?31?32??0?3?34?51?????? ?2?34?527??0?741?25?????9?96162250?27611?1616????0?32?1?120?32?1?12?0?3?3?1?0?14?5??334?5???? ???2529?8?2529? 81?0011?3??001??33333????0033?252970000?01????????因为
rank(A)?4?rank(A)?3
所以原方程无解.
3）对方程组德增广矩阵作行初等变换，有
?1?23?44??1?23?44??01?11?3??01?11?3?????? ?13011??05?35?3?????0?731?30?731?3????
?1?0???0??0因为
01?2?2??1?01?11?3????02012??0??0?48?24??0010000200?8?03?? 012??80?rank(A)?rank(A)?4
所以方程组有惟一解，且其解为
?x1??8?x?3?2 ??x3?6??x4?04）对方程组的增广矩阵作行初等变换，有
?34?57??17?89??2?33?2??2?33?2?????? ?411?1316??411?1316?????7?2137?213?????89??17?89??17?0?1719?20??0?1719?20????? ???017?1920??0000?????0?3438?400000????即原方程组德同解方程组为
?x1?7x2?8x3?9x4?0 ??17x?19x?20x?0234?由此可解得
313?x?k??117117k2??x?19k?20k?2171172 ??x3?k1?x?k?42其中k1,k2是任意常数g
5）对方程组的增广矩阵作行初等变换，有
?21?111??2?3?22?32??7?????51?12?1??3???2?11?34???4?2?7???10???10因为
1?111?00?14?? 001?2??00?25?1?111?00?14?? 0002??000?1?1?111??2?700?14????0002??10??000?3??0rank(A)?4?rank(A)?3
所以原方程组无解.
6）对方程组的增广矩阵作行初等变换，有
?1?3? ?2??2??5??2?5?????1???1?0?232131225205221??3?2?11???11???2???11??402????50015402?5202??3111?
?5302?5202??050?2??61?0?1??
55?0100?0000??07000??0?02???11??0?1????055??0100???1?0000???0即原方程组的同解方程组为
?5x2?7x3?2?61? ?x?x???345?5???x1?x3?0解之得
?x1?k??x2?2?7k?55 ??x3?k?16x???k?455?其中k是任意常数.
20.求下列齐次线性方程组的一个基础解系，并用它表出全部解：
?x1?x2?x3?x4?x5?0?x1?x2?3x4?x5?0?3x?2x?x?x?x?0?x?x?2x?x?0?1?122345341）?2）?
x?2x?2x?6x?04x?2x?6x?3x?4x?03452345?2?1???5x1?4x2?3x3?3x4?x5?0?2x1?4x2?2x3?4x4?7x5?0?x1?2x2?x3?x4?x5?0?x1?2x2?x3?x4?x5?0?2x?x?x?x?x?0?2x?x?x?2x?3x?0?12345?12345 3）?4）?
x?7x?5x?5x?5x?03x?2x?x?x?2x?023452345?1?1???3x1?x2?2x3?x4?x5?0?2x1?5x2?x3?2x4?2x5?0解 1）对方程组的系数矩阵作行初等变换，有
?1?3??0??5r1214112311??11111??11111??0?1?2?2?6??0?1?2?2?6?1?3????????26??01226??00000??????3?1??0?1?2?2?6??00000?因为
(a?n)?，所以原方程组的基础解中含有kA3个线性无关的解向量，且原方程组的同
解方程组为
?x1?x2?x3?x4?x5?0 ??x2?2x3?2x4?6x5?0于是只要令
x3?1,x4?x5?0, 即得?1?(1,?2,1,0,0)?
同理，令
x1?1,x3?x5?0, 即得?2?(1,?2,0,1,0)? x5?1,x3?x4?0, 即得?3?(5,?6,0,0,1)?
则?1,?2,?3为原方程组的一个基础解系，且该齐次线性方程组的全部解为
??k1?1?k2?2?k3?3
其中k1,k2,k3为任意常数.
2）对方程组的系数矩阵作行初等变换，有
?110?3?1??110?3?1??1?12?10??0?2221?????? ?4?263?4??0?66150?????24?24?702?210?5?????11?0?2???00??000?3?1??11?0?2221????09?3??00??012?4??000?3?1?221?? 03?1??000?因为rank(A)?3?5，所以原方程组的基础解系中含有2个线性无关的解向量，且原方程组的同解方程组为
?x1?x2?x4?x5?0???2x2?2x3?2x4?x5?0 ?3x?x?0?45若令
x1?1,x4?0，得?1?(?1,1,1,0,0)?
x1?0,x4?1，得?2?(?75,,0,1,3)? 22则?1,?2为原方程组的一个基础解系，且该齐次线性方程组的全部解为
??k1?1?k2?2
其中k1,k2为任意常数.
3）对方程组的系数矩阵作行初等变换，有
?1?211?1??1?211?1??21?1?1?1??05?3?31?????? ?17?5?55??09?6?66?????3?1?21?105?5?22????11?1??1?211?2??1?2?05?13?31??05?3?31????? ???0?2112120??0?6900?????140??0?5140??0?5又因为
111?20?3?31?0
09000140所以rank(A)?4?5，方程组的基础解系含有一个线性无关的解向量，且原方程组的同解方程组为
?x1?2x2?x3?x4?2x5?0?5x?3x?3x?x?0?2345 ??6x?9x?023????5x2?x3?4x4?0于是令x2?1，可得
??(,1,,122131,)? 3124则?即为原方程组的一个基础解系，且该齐次线性方程组的全部解为k?，其中k为任意常数.
4） 对方程组的系数矩阵作行初等变换，有