线性代数第二版答案

线性代数第二版答案 1 2 3 4 5 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? ?x2?315?y2? ?x?323?y?2?3? ?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 ?1 ?y1?3z1?z2?x1?2y1?y3 ?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 ?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? 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?21322? ?3?0?56?2?11?1?2?1720? ?290?1?11?429?2? ?111?123?058? ATB?11?1?1?24?0?56? ?1?11?051?290? 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?3?2?2?3?1)?(10)?1? ?2? (3)?1?(?12)? ?3? ?2?(?1)2?2?24?2? 解 ?1?(?12)?1?(?1)1?2?12? ?3?3?(?1)3?2?36? ?131?0?12?2140? (4)?1?31? ? 1?134?40?2? ?131?0?12?6?78?2140? 解 ?1?31?20?5?6? 1?134?40?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? 5? 设A?22?a11x12?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3? ?1 ?12? B?1?13?0? 问? 2? (1)AB?BA吗? 解 AB?BA? 因为AB?3 ?44? BA?1?36?2? 所以AB?BA? 8? (2)(A?B)2?A2?2AB?B2吗? 解 (A?B)2?A2?2AB?B2? 因为A?B? ?2?22? 5?2?2?25?2(A?B)2?2? 但2?814? ?1429?5?38?68?1A2?2AB?B2?411?812?3?0?1016? ?1527?4?所以(A?B)2?A2?2AB?B2? (3)(A?B)(A?B)?A2?B2吗? 解 (A?B)(A?B)?A2?B2? 因为A?B? ?2?22? A?B?0?05?2? 1?22?02?06? (A?B)(A?B)?25?01?09? 10?28? ?38?而A2?B2?411?34?17? 故(A?B)(A?B)?A2?B2? 6? 举反列说明下列命题是错误的?（也可参考书上的答案） (1)若A2?0? 则A?0? 解 取A?0 ?0 ?1 ?01? 则A2?0? 但A?0? 0?1? 则A2?A? 但A?0且A?E? 0? (2)若A2?A? 则A?0或A?E?解 取A? (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 ? 7? 设A? 解 ?10? 求A2? A3? ? ? ? Ak? ?1?10?10?10? A2?1?1?2?1?10?10?10? A3?A2A?2?1?1?3?1? 0? 1? ? ? ? ? ? ? 1Ak?k? ?10? 8? 设A?0?1? 求Ak ? ?00? 解 首先观察 ?10?1A2?0?1?0?00?00?33?2A3?A2?A?0?3?00?44?3A4?A3?A?0?4?00?55?4A5?A4?A?0?5 ?00? ?k?kA?0?0k?k