同济大学线性代数第五版课后习题答案

51 D?000651000651000651000?66565

10 D1?00151 D3?000651006510006510100010065100651000?150765000?7036551 D2?00051 D4?000100016510006510065100065110001000??114565000??39565

51 D5?000651000651000651100?21201

所以 x1?1507665 x2??1145665

x3?703665取何值时

x4??395665 x4?212665

9 问 齐次线性方程组

???x1?x2?x3?0?x1??x2?x3?0有非零解？ ??x1?2?x2?x3?0 解 系数行列式为

?11 D?1?1?????12?1 令D0 于是 10

问 当

得 0或0或

1

1时该齐次线性方程组有非零解

取何值时 齐次线性方程组

??(1??)x1?2x2?4x3?0?2x1?(3??)x2?x3?0有非零解？ ??x1?x2?(1??)x3?0 解 系数行列式为

1???241???3??4 D?23??1?21??1

111??101?? (1

)

3

((1 得 0

3))

3

4(12(1

)2(1)

2

)(33

)

令D0 于是零解

当

2或2或

3

该齐次线性方程组有非

0 3时

第二章 矩阵及其运算

1 已知线性变换

??x1?2y1?2y2?y3?x2?3y1?y2?5y3??x3?3y1?2y2?3y3求从变量x1

x2

x3到变量y1 y2 y3的线性变换

解 由已知

?x1??221??y1? ?x2???315??y2??x??323??y???2??3???1?y1??221??x1???7?49??y1?故 ?y2???315??x2???63?7??y2??y??323??x??32?4?????3????y3??2??

??y1??7x1?4x2?9x3 ?y2?6x1?3x2?7x3??y3?3x1?2x2?4x3 2 已知两个线性变换

??x1?2y1?y3 ?x2??2y1?3y2?2y3??x3?4y1?y2?5y3??y1??3z1?z2 ?y2?2z1?z3??y3??z2?3z3

求从z1 z2 z3到x1 x2 x3的线性变换

解 由已知

??x0?x1??201??y1??201???312?1????x3???????42312??5???y2????232??20?y???415?????z1?2??0?13??z2??z? 3? ????613??z?12?49??z1?2?

??10?116????z?3?所以有??x1??6

?x?10zz1?z2?3z?32?121z?4z2?9z3?x3?1?z2?16z3 3 设A???111?1?1?11?1? B???123?3AB?1????01?524?1? 求??ATB

解 3AB?2A?3??111??123??111??11?1???1?24??2?11?1??1?11?

???051????1?11?? ?3??058??111???213?0?56??2?11?1????2?172220??

?290????1?11????429?2?? ATB???111??123??11?1???1?24????00?5586?? ?1?11????051????290?? 4 计算下列乘积

2A及