?1??1??6??7??7??x??7??9??5??y??5?即 ???k1???k2???????(k1
z7??7????w??700???1??1??0??0????? 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 ?2x3?c1?x?c?42
x1?2x3?x4或 ??x??3x?4x?234
x1?2x3?x4?0或 ??x?3x?4x?0?234 15
取何值时
这就是一个满足题目要求的齐次线性方程组
非齐次线性方程组
???x1?x2?x3?1?x1??x2?x3??2?x?x??x???123
(1)有唯一解 (2)无解 (3)有无穷多个解?
解 B????111??1?1??
?11??2?? ~r ??1?0?1?11????(?2 ?1??)?00(1?)(2??)(1??)(??1)2?? (1)要使方程组有唯一解 必须R(A)
3 因此当
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)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??
?x1?x2?2x3??2当取何值时有解？并求出它的解
1
3
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
2
当
1时
必须(1
)(
2)
0 即
1
??211?2??10?11? B??1?211?~?01?10??11?21??0000?????方程组解为
?x1?x3?1x1?x3?1?? ?或?x2?x3x?x?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?????
17问
??(2??)x1?2x2?2x3?1 设?2x1?(5??)x2?4x3?2???2x1?4x2?(5??)x3????1
为何值时 此方程组有唯一解、无解或有无穷多解? 并在
有无穷多解时求解 解 B2?21??2???25???42? ??2?45?????1???
?42?25????? ~01??1??1???00(1??)(10??)(1??)(4??)??? 要使方程组有唯一解 (1所以当 (1所以当 (1所以当
1时1且
)(10
10时)(1010时
必须R(A))0
方程组有唯一解. 必须R(A)R(B))0且(1
)(4
R(B)3 即必须
要使方程组无解 即必须
)
0
方程组无解.
必须R(A)
)(4
要使方程组有无穷多解
)(10
R(B)3 即必须
)
0
)0且(1
方程组有无穷多解
此时，增广矩阵为
?12?21? B~?0000??0000???方程组的解为 ??x1??x2?x3?1 ?x2? x2??x3? x3
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