于是零解
当
0 2或3时 该齐次线性方程组有非
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
解 由已知
x3到变量y1 y2 y3的线性变换
?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?y ?z??y?1??3 ?x3x2??2y1?3y2?2y3?3?4y1?y2?5y3?y1?z22?2z1?z3?y3??z2?3z3求从z1
z2
z3到x1
x2
x3的线性变换
解 由已知
??x1??201??y10??x??201???31???z11??x2?3???????42312??5?y2??????232??20?y??2??415????0?13???z2? ?z?3? ????613??z1??
??1210??41169?????z?z2??3?所以有??x1??6z1?z2?3z?x32?12z1?4z2
???10z?916z3?x31?z2?z3 3 设A???1?111?11?? B???123?3AB?1?11????01?524?1 求???ATB
解 3AB?2A?3??1?11?1?11??123??111?1?11???????01?524?1??2???1?1?11?11??
? ?3??0?0?5586???2??1111???21322?
?290?1?1????2?1720????1?11????429?2?? ATB???1?111?11?????11?2234???058?
??11????051???0?56?1???290?? 4 计算下列乘积
2A及
?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???
(132231)(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? (a11x1a12x2a13x3
?x1?a33x3)?x2?
?x??3? a12x1a22x2a23x3
a13x1a23x2
22 ?a11x12?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3
1 5 设A???1? 解 AB2?3??1 B???1?0?2?? 问
(1)ABBA吗?
BA
4?6??1 BA???3?2?8?? 所以AB3 因为AB???4? (2)(AB) 解 (A2
BA
A22ABB2吗?
2?5??B)2A22ABB2
2 因为A?B???2?2 (A?B)2???2?2??2?25???2???814??1429?5????
38??68???1但 A2?2AB?B2???411???812??3?????所以(A0???1016??1527?4????
B)2A22ABB2
B)A2B2吗? B)(AB)A2B2
(3)(AB)(A 解 (A
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