线性代数第五版答案(全)(5)

? ?x?x1?02?0?

?x3?0?x4?0?3x (4)?2x1?4x?3x2?5x3?7x4?0??4x12?3x3?2x4?0?

1?11x2?13x3?16x?7x1?2x2?x3?3x4?04?0 解 对系数矩阵A进行初等行变换? 有

A????43?53?7?10?313?3?2???~?1701?1917?220??411?1316???? ?7?213???1717?0?000000?0??

??x1?317x3?1317x4于是 ??x19202?x3?x?17174?

?x?xx3?x3?44故方程组的解为

??3?? ?x??13?x1??17?17??2??19??20??x??k31???k2??17?(k1? k2为任意常数)??x??17???4??10?0????1??

13? 求解下列非齐次线性方程组:

??4x1?2x2?xx3?2 (1)?3x?1?1x2?2?83?10?

?11x1?3x2 解 对增广矩阵B进行初等行变换? 有

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B???4?3?21?21102??~??11308????010?103?0113?0?348?6???

于是R(A)?2? 而R(B)?3? 故方程组无解?

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

?4x?y?9z??6 解 对增广矩阵B进行初等行变换? 有

? B??2?1?3241?45????0101?21?21???38?213?~?0000??

?4?19?6????0000??于是 ??x??2z?1??y?z?2z?z?

?即 ??x???2???1??y??k?1???2?(k为任意常数)??z????1????0?

???2x?y?z?w?1 (3)??4x?2y?2z?w?22x?y?z?w?1?

? 解 对增广矩阵B进行初等行变换? 有

B???21?111??11/2?1/2?40011/20??? ?221??21?112?1?~?00???00000???x??1y于是 ???y?y2?12z?12? ?z?z

??w?0?x???1??1??1?即 ??y??????k2?2??2?1?1??k2?0???0?(k1? k?wz????0??0????1??2为任意常数)?

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

? 解 对增广矩阵B进行初等行变换? 有

B???21?111??3?21?34?~?10?1/7?1/76/7??14?35?2??01?5/79/7?5/7??00000??? 34

??x?1z?1w?6于是 ?777?y?5z?9w?5?

?777?z?z?w?w??1???17????67??即 ?x?y?????k?7?5??k?9??5??1??2???????(k1? ?wz???7?1??7k2为任意常数)?

??7?000?????1????0?? 14? 写出一个以

?x?c??23?????24??1????c2??

?01?0???1??为通解的齐次线性方程组? 解 根据已知? 可得

?x1??2? ??x2??x??c??3??c??24??31????x??2???

?01?0??4??1??与此等价地可以写成

?x ??x1?2c2??31?cc?214c2? ?x?c

31?x4?c2或 ??x1?2x3?x4?x2??3x3?4x?

4或 ??x1?2x3?x4?0?x2?3x3?4x4?0? 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?x???x3?1?x12?x3???

?x??21?x2??x3 (1)有唯一解? (2)无解? (3)有无穷多个解?

解 B????111??1?1???11??2?

? 35

211???1???(1??)? ~ ?0??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?x?2x3??2?x?12?x3??

?x?2x21?x23??当?取何值时有解？并求出它的解?

解 B????211?2???1??1?21???2(??1)??1?21?21??2?~??01?1??? ?000(??13)(??2)??要使方程组有解? 必须(1??)(??2)?0? 即??1? ???2? 当??1时?

B????2?1?2111?21???10?11??11?21?~?01?10??

???0000??方程组解为

x ?1?x3?1?x1?x?1???x?x3或?xx2?x23?3?

?3?x3即 ??x?x1??1??1?2??k?1???0?(k为任意常数)?

?x??1??3????0?? 当???2时?

B????2?1?2111??22??~??0101??1122????11?24?

???0000??方程组解为

??x1?x3?2??x1?x3?2?x或2?x3?2?x?2?x3?2?

?x3?x3

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?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? 并在有无穷多解时求解?

2?21??2??2? 解 B??25???4??2?45?????1????42?25?????? 1??1?? ~01???00(1??)(10??)(1??)(4??)??? 要使方程组有唯一解? 必须R(A)?R(B)?3? 即必须 (1??)(10??)?0?

所以当??1且??10时? 方程组有唯一解.

要使方程组无解? 必须R(A)?R(B)? 即必须 (1??)(10??)?0且(1??)(4??)?0? 所以当??10时? 方程组无解.

要使方程组有无穷多解? 必须R(A)?R(B)?3? 即必须 (1??)(10??)?0且(1??)(4??)?0?

所以当??1时? 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?x1???2??2??1?或 ?x2??k1?1??k2?0???0?(k1? k2为任意常数)?

?x??0??1??0??????3??? 18? 证明R(A)?1的充分必要条件是存在非零列向量a及非零行向量bT? 使A?abT?

证明 必要性? 由R(A)?1知A的标准形为

?10???0??1??00???0??0??(1, 0, ???, 0)? ????????????????????00???0??0????? 37

即存在可逆矩阵P和Q? 使

?1??1??0????10 PAQ???(1, 0, ???, 0)? 或A?P??(1, 0, ???, 0)Q?1? ???????0??0??????1???T?10 令a?P??? b?(1? 0? ???? 0)Q?1? 则a是非零列向量? bT是非零行向量? 且A?abT? ????0??? 充分性? 因为a与bT是都是非零向量? 所以A是非零矩阵? 从而R(A)?1? 因为

1?R(A)?R(abT)?min{R(a)? R(bT)}?min{1? 1}?1? 所以R(A)?1?

19? 设A为m?n矩阵? 证明

(1)方程AX?Em有解的充分必要条件是R(A)?m?

证明 由定理7? 方程AX?Em有解的充分必要条件是

R(A)?R(A? Em)?

而| Em|是矩阵(A? Em)的最高阶非零子式? 故R(A)?R(A? Em)?m? 因此? 方程AX?Em有解的充分必要条件是R(A)?m?

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