线性代数课后答案（高等教育出版社）

?a11a12a13??x1? (x1x2x3)?a12a22a23??x2?

????aaa?132333??x3??x1? ?(a11x1?a12x2?a13x3 a12x1?a22x2?a23x3 a13x1?a23x2?a33x3)?x2?

?x??3?222 ?a11x1?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3?

?111??123?T

2. 设A??11?1?? B???1?24?? 求3AB?2A及AB?

?1?11??051??????111??123??111? 解 3AB?2A?3?11?1???1?24??2?11?1?

?1?11??051??1?11????????058??111???21322? ?3?0?56??2?11?1????2?1720??

?290??1?11??429?2????????111??123??058? AB??11?1???1?24???0?56??

?1?11??051??290???????T3. 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?0??z1?1??z2? ?z?3???3???x1??6z1?z2?3z3所以有?x2?12z1?4z2?9z3?

??x3??10z1?z2?16z34.

1设A???1?2?? B??1?13???0?? 问? 2?? (1)AB?BA吗? 解 AB?BA?

3 因为AB???4?

4?? BA??1?36???2?? 所以AB?BA? 8?? (3)(A?B)(A?B)?A2?B2吗? 解 (A?B)(A?B)?A2?B2?

2 因为A?B???2?2?? A?B??0?05???2??0?05???2??

1??6??

9??2 (A?B)(A?B)???2?2???0?01???38???10???2而 A2?B2???411??34??1?????故(A?B)(A?B)?A2?B2?

5. 举反列说明下列命题是错误的? (1)若A2?0? 则A?0?

8??

7??0 解 取A???0?1?? 则A2?0? 但A?0? 0?? (2)若A2?A? 则A?0或A?E?

1 解 取A???0? 解 取

1?? 则A2?A? 但A?0且A?E? 0?? (3)若AX?AY? 且A?0? 则X?Y ?

10?? X??11?? Y??11?? A???00???11??01???????则AX?AY? 且A?0? 但X?Y ?

??10?7. 设A??0?1?? 求Ak ?

?00???? 解 首先观察

??10???10???22?1? A2??0?1??0?1???0?22???

?00???00???00?2?????????33?23?? A3?A2?A??0?33?2??

?00?3?????44?36?2? A4?A3?A??0?44?3??

?00?4?????55?410?3? A5?A4?A??0?55?4??

?00?5??? ? ? ? ? ? ??

??kk?k?1k(k?1)?k?2?2k A??0?kk?k?1?00?k???? ? ?? 用数学归纳法证明? 当k?2时? 显然成立? 假设k时成立，则k?1时，

??kk?k?1k(k?1)?k?2?????10?2 Ak?1?Ak?A??0?kk?k?1??0?1?

?00??00???k????????k?1(k?1)?k?1(k?1)k?k?1???2?k?1(k?1)?k?1?? ??0??k?100?????由数学归纳法原理知?

??kk?k?1k(k?1)?k?2???2 Ak??0?kk?k?1??

?00??k????

8. 设A? B为n阶矩阵，且A为对称矩阵，证明BTAB也是对称矩阵?

证明 因为AT?A? 所以

(BTAB)T?BT(BTA)T?BTATB?BTAB? 从而BTAB是对称矩阵? 11? 求下列矩阵的逆矩阵?

1 (1)??2?2?? 5??12?? |A|?1? 故A?1存在? 因为 解 A???25???A11A21??5?2? A*???AA????21??

??1222??5?2? 故 A?1?1A*????21??|A|???12?1?(3)?34?2?? ?5?41????12?1? 解 A??34?2?? |A|?2?0? 故A?1存在? 因为

?5?41????A11A21A31???420? A*??A12A22A32????136?1??

????3214?2?AAA??132333??