51 D3?00051 D5?000651006510010001065100065100651000?70365100?2120151 D4?000651000651010001000??39565
所以
x1?1507? x2??1145? x3?703? x4??395? x4?212?
665665665665665???x1?x2?x3?0 9? 问?? ?取何值时? 齐次线性方程组?x1??x2?x3?0有非
??x1?2?x2?x3?0零解？
解 系数行列式为
?11 D?1?1?????12?1 令D?0? 得 ??0或??1
于是 当??0或??1时该齐次线性方程组有非零解?
??(1??)x1?2x2?4x3?0 10? 问?取何值时? 齐次线性方程组?2x1?(3??)x2?x3?0有
??x1?x2?(1??)x3?0非零解？
解 系数行列式为
1???241???3??4 D?23??1?21??1
111??101?? ?(1??)3?(??3)?4(1??)?2(1??)(?3??) ?(1??)3?2(1??)2???3? 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
解 由已知? ?x1??221??y1? ?x2???315??y2??x??323??y???2??3??
y3的线性变换?
?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求从z1
z2
z3到x1
x2
解 由已知
??y1??3z1?z2 ?y2?2z1?z3??y3??z2?3z3
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?16z3
?111??123? 3? 设A??11?1?? B???1?24??1?11??051????? 求3AB?2A及ATB
?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???????T
4? 计算下列乘积?
?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
?1???3?22?31)?(10)
?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)???11?134????43?1?301?2? ? 1??2???1?02140?? 解 ???11?134????43?1?301?2???6?78??20?5?61??????2?
?a11a12a13??x1? (5)(x1x2x3)?a12a22a23??x2??
????aaa?132333??x3? 解
?a11a12a13??x1? (x1x2x3)?a12a22a23??x2?
????aaa?132333??x3??x1? ?(a11x1?a12x2?a13x3 a12x1?a22x2?a23x3 a13x1?a23x2?a33x3)?x2?
?x??3?22 ?a11x12?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3
1 5? 设A???1?2?? B??1?13???
0?2?? 问?
(1)AB?BA吗? 解 AB?BA
3 因为AB???4?4?6??1 BA???3?2?8??
所以AB?BA
(2)(A?B)2?A2?2AB?B2吗? 解 (A?B)2?A2?2AB?B2
2 因为A?B???2?2?5??
工程数学线性代数(同济大学第六版)课后习题答案(全)
