线性代数课后答案(高等教育出版社)

?1?3 (3)?2?3??1?3?2?33534?4?4?2?235343?1?? 0??1??3?1?(下一步? r?3r? r?2r? r?3r? )

213141

0??1???1?3 解 ?2?3??1?3?2?3?4?4?2?2?1?0 ~?0?0??1?0 ~?0?0??1?0 ~?0?0??13?43?0?48?8?(下一步? r?(?4)? r?(?3) ? r?(?5)? )

234

0?36?6?0?510?10???1000?10003111?4?2?2?23?2?(下一步? r?3r? r?r? r?r? )

123242

2?2??02?3?1?22?? 000?000??3. 已知两个线性变换

??x1?2y1?y3 ?x2??2y1?3y2?2y3?

??x3?4y1?y2?5y3 解 由已知

??y1??3z1?z2?y2?2z1?z3? ??y3??z2?3z3求从z1? z2? z3到x1? x2? x3的线性变换?

?x1??201??y1??201???31 ?x2????232??y2????232??20?x??415??y??415??0?1??2?????3????613??z1? ??12?49??z2??

??10?116??z????3???x1??6z1?z2?3z3所以有?x2?12z1?4z2?9z3?

??x3??10z1?z2?16z3

0??z1?1??z2? ?z?3???3?4. 试利用矩阵的初等变换? 求下列方阵的逆矩阵?

?321? (1)?315??

?323????321100??321100? 解 ?315010?~?0?14?110?

?323001??002?101??????3203/20?1/2??3007/22?9/2? ~?0?1011?2?~?0?1011?2?

?002?10??1????001?1/201/2??1007/62/3?3/2? ~?010?1?12?

?001?1/201/2????72?3??632?

故逆矩阵为??1?12??

?11??0?2?2???3?20?1??0221? (2)??

1?2?3?2??0121???

?3?20?11000??02210100? 解 ?

1?2?3?20010??01210001????1?2?3?20010??01210001? ~?

049510?30??02210100????1?2?3?20010??01210001? ~?

001110?3?4??00?2?1010?2????1?2?3?20010??0121000?1 ~? 001110?3?4??000121?6?10????1?200?0100 ~?0010?0001??1?0 ~?0?0?01000010?10?12?11`?11?2?2?0?1? 36??6?10???1?0故逆矩阵为??1?2?011?2?4?0010?1? 0?1?136?121?6?10??1?2?4?10?1?? ?136?1?6?10???021?123?? 求X使XA?B? 5. (2)设A??2?13?? B????2?31??33?4????? 解 考虑ATXT?BT? 因为

?02?312?r?1002?4? (AT, BT)??2?132?3?~ ?010?17??

?13?431??001?14??????2?4?所以 XT?(AT)?1BT???17??

??14???2?1?1? 从而 X?BA?1????474????9. 求作一个秩是4的方阵? 它的两个行向量是

(1? 0? 1? 0? 0)? (1? ?1? 0? 0? 0)?

解 用已知向量容易构成一个有4个非零行的5阶下三角矩阵?

?1?1?1?0?0?0?100000100000100?0?0?? 0?0??此矩阵的秩为4? 其第2行和第3行是已知向量?

?1?23k?12. 设A???12k?3?? 问k为何值? 可使

?k?23??? (1)R(A)?1? (2)R(A)?2? (3)R(A)?3?

k?1?23k?r?1?1?????? 解 A??12k?3~ 0k?1k?1?k?23??00?(k?1)(k?2)????? (1)当k?1时? R(A)?1? (2)当k??2且k?1时? R(A)?2? (3)当k?1且k??2时? R(A)?3? P106/ 1.已知向量组

A? a1?(0? 1? 2? 3)T? a2?(3? 0? 1? 2)T? a3?(2? 3? 0? 1)T? B? b1?(2? 1? 1? 2)T? b2?(0? ?2? 1? 1)T? b3?(4? 4? 1? 3)T? 证明B组能由A组线性表示? 但A组不能由B组线性表示? ?0?1 证明 由 (A, B)??2?3??1r? ~ ?00?0?031?60200430122301204??1r?1?24?~0 111??0?213???0031?24?32204? 1?6?15?7?2?8?17?9??1?24??1r??15?7?~0 5?1525??0?1?35???0031?24?1?6?15?7? 041?35?00000??知R(A)?R(A? B)?3? 所以B组能由A组线性表示? 由