线性代数习题解答(同济大学（第四版）)(7)

二阶子式

31??4．

1?1r1?r2?2??13?4?41??r2?2r1???3?~?0?7119?5?

??7??8??r3r1?0?213327?15??41??9?5?秩为2. 00???32?1?3?(2) ?2?131?705?1??13?4?r3?3r2?0?711~?0?0032二阶子式??7．

2?12?17??21837??01r?2r??1?4??2?307?5??0?3?63?5?(3)

?3?2580?~?0?2?42?0r2?2r4?????10320??10320???r3?3r4??r1?r2?012?17??10320?r2?3r1??r4?r1???000016??012?17?~?000014?~?00001?秩为3

r3?14?r3?2r1???10320??00000????r4?16??r4?r307?5580??5?70?0． 三阶子式5832320

6．求解下列齐次线性方程组:

?x1?x2?2x3?x4?0,?x1?2x2?x3?x4?0,??(1) ?2x1?x2?x3?x4?0, (2) ?3x1?6x2?x3?3x4?0,

?2x?2x?x?2x?0;?5x?10x?x?5x?0;234234?1?1?2x1?3x2?x3?5x4?0,?3x1?4x2?5x3?7x4?0,?3x?x?2x?7x?0,?2x?3x?3x?2x?0,?1?1234234(3) ? (4)?

?4x1?x2?3x3?6x4?0,?4x1?11x2?13x3?16x4?0,???x1?2x2?4x3?7x4?0;?7x1?2x2?x3?3x4?0.解 (1) 对系数矩阵实施行变换:

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4?x?x41?3??112?1????10?10?????x2??3x4? ?211?1?~?013?1?即得??2212??001?4??x?4x???3343???x?x?44?4?x?1?????3???x2???3? ?k故方程组的解为

?x??4?3??3??x???1??4???

(2) 对系数矩阵实施行变换:

?121?1??120?1??????36?1?3?~?0010??5101?5??0000????? ?x1???2??1????????x2??1??0?故方程组的解为

?x??k1?0??k2?0?

3????0???1???x???????4?

(3) 对系数矩阵实施行变换:

?x1??2x2?x4?x?x?22即得?

?x3?0??x4?x4?23?15??1???2?7??0?31?41?36?~?0??1?24?7???????0?x1?0?x?0?2故方程组的解为?

?x3?0??x4?0

(4) 对系数矩阵实施行变换:

01000010?x1?00???x?00??2即得? ?0?x3?0??1???x4?032

313??10???34?57??1717????19202?33?2???01??~?411?1316??1717????0000??7?21?3??? ?0??000313?x?x??117317x4??x?19x?20x即得?234

1717??x3?x3?x?x?44?3??13???????x1????17??17?19?20??x2???故方程组的解为

?x??k1?17??k2??17?

3??1??0??x???????4??0??1?

7．求解下列非齐次线性方程组:

?2x?3y?z?4,?4x1?2x2?x3?2,?x?2y?4z??5,??(1) ?3x1?1x2?2x3?10, (2) ?

?11x?3x?8;?3x?8y?2z?13,?12??4x?y?9z??6;?2x?y?z?w?1,?2x?y?z?w?1,??(3) ?4x?2y?2z?w?2, (4) ?3x?2y?z?3w?4,

?2x?y?z?w?1;?x?4y?3z?5w??2;??

解 (1) 对系数的增广矩阵施行行变换,有

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

??11308??000?6????R(A)?2而R(B)?3,故方程组无解．

(2) 对系数的增广矩阵施行行变换:

14??1?23????1?24?5??0?38?213?~?0??4?19?6????0???02?1??1?12? ?000?000??33

?x??2z?1?x???2???1????????即得?y?z?2亦即?y??k?1???2?

?z??1??0??z?z???????

(3) 对系数的增广矩阵施行行变换:

?21?111??21?111??????42?212?~?00010? ?21?1?11??00000?????111??1??1??1?xx??y?z?????????????22222??2?????y???1??k?0???0? ??k即得?y?y 即12?z??0??1??0??z?z???w????????????????000?w?0???????

(4) 对系数的增广矩阵施行行变换:

1??14?35?2??21?11???595????3?21?34?~?01?777??14?35?2?????00000?116??10????777??595~?01??? ?777??00000?????116??1??1??6?x?z?w?????????x?777777?????????5?9??5??y?5z?9w?5?y?????k?k???? 即得? 即1??2????777777z????1??0??0???z?z????????w??w?w010???????

8．?取何值时,非齐次线性方程组

??x1?x2?x3?1,??x1??x2?x3??, ?2x?x??x??23?1(1)有唯一解；(2)无解；(3)有无穷多个解?

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解 (1)

?111?1?0,即??1,?2时方程组有唯一解. 11?

R(A)?R(B)

1??2??111??1???1???(1??)? B??1?1??~?0??12??11??2??00(1??)(2??)(1??)(??1)????2由(1??)(2??)?0,(1??)(1??)?0 得???2时,方程组无解.

(2)

(3)

R(A)?R(B)?3,由(1??)(2??)?(1??)(1??)2?0,

得??1时,方程组有无穷多个解.

9．非齐次线性方程组

当?取何值时有解？并求出它的解．

??2x1?x2?x3??2,??x1?2x2?x3??, ?2x?x?2x??23?1?1?21?1?2????21????2?21??~?01?1?(??1)? 解 B??13??2??1?1?2?0(??1)(??2)????00?方程组有解，须(1??)(??2)?0得??1,???2

?x1??1??1???????当??1时,方程组解为?x2??k?1???0?

?x??1??0??3??????x1??1??2???????当???2时,方程组解为?x2??k?1???2?

?x??1??0??3?????

?(2??)x1?2x2?2x3?1,?10．设?2x1?(5??)x2?4x3?2,

??2x?4x?(5??)x????1,123?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．

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