令D?0?得
??0???2或??3?
于是?当??0???2或??3时?该齐次线性方程组有非零解?
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??x???315??y?? ?x2??323??y2???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?4x32?已知两个线性变换
???x1?2y1?y3?y1??3z1?z2?x2??2y1?3y2?2y3??y2?2z1?z3?
???y3??z2?3z3?x3?4y1?y2?5y3求从z1?z2?z3到x1?x2?x3的线性变换? 解由已知
??613??z1???12?49??z2?? ??10?116??z????3???x1??6z1?z2?3z3所以有?x2?12z1?4z2?9z3?
??x3??10z1?z2?16z3?111??123?3?设A??11?1??B???1?24??求3AB?2A及ATB?
?1?11??051??????111??123??111?解3AB?2A?3?11?1???1?24??2?11?1?
?1?11??051??1?11????????058??111???2?3?0?56??2?11?1????2?290??1?11??4??????111??123??0ATB??11?1???1?24???0?1?11??051??2?????4?计算下列乘积?
1322??1720?? 29?2??58??56?? 90???431??7?(1)?1?23??2?? ?570??1??????431??7??4?7?3?2?1?1??35?解?1?23??2???1?7?(?2)?2?3?1???6?? ?570??1??5?7?7?2?0?1??49??????????3?(2)(123)?2??
?1????3?解(123)?2??(1?3?2?2?3?1)?(10)?
?1????2?(3)?1?(?12)? ?3????2?(?1)2?2???2?2?解?1?(?12)??1?(?1)1?2????1?3??3?(?1)3?2???3?????4?2?? 6???1?02140??(4)????1?134??1?4?1?02140??解???11?134????43?1?303?1?301?2?? 1??2??1?2???6?78??
?20?5?61??????2??a11a12a13??x1?(5)(x1x2x3)?a12a22a23??x2??
????aaa?132333??x3?解
?x1??(a11x1?a12x2?a13x3 a12x1?a22x2?a23x3 a13x1?a23x2?a33x3)?x2?
?x??3?22?a11x12?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3?
15?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??(2)(A?B)2?A2?2AB?B2吗? 解(A?B)2?A2?2AB?B2?
2因为A?B???2?2?? 5??2(A?B)2???2?2??2?25???2???814??
?1429?5????38???68???10???1016?? 但A2?2AB?B2???411??812??34??1527?????????所以(A?B)2?A2?2AB?B2? (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?5???02??
1??6?? 9??2(A?B)(A?B)???2?2???0?1???038??10???2而A2?B2???411???34??1?????故(A?B)(A?B)?A2?B2?
8??
7??6?举反列说明下列命题是错误的? (1)若A2?0?则A?0?
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?1??则A2?A?但A?0且A?E? 0??(3)若AX?AY?且A?0?则X?Y? 解取
1A???0?0??X??11??Y??1??11??00?????1??
1??则AX?AY?且A?0?但X?Y?
10?求A2? A3?????Ak? 7?设A????1????10?10???10?? 解A2????1????1??2?1???????10?10???10?? A3?A2A???2?1????1??3?1??????????????
10? Ak???k?1??????10?8?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???
同济大学工程数学线性代数第六版答案全
