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1、第一章第一章 矩阵矩阵 。11, yxyx,cossinsincos)2(000) 1 (1111yxyyxxyxyyxx1、写出下列由变量到变量的线性变换的 系数矩阵：0001AcossinsincosA2、二省城市间如下图，每条线上的数字表示 连接这两个城市的不同通路总数，试用矩阵形式表示城市间的通路情况。1a2a1b3b2b1a2a1b2b3b2203141b2b3b1a2a23210422413 ,111111111A,150421321BAAB23BAT229420172221320926508503.4、计算2210013112) 1 (11)2(212222111211yxcbb
2、baabaayx433349447ybxbxyacyaxa2112222211222ybxbxyacyaxa12122121122222211zyxaaaaaaaaazyx333231232221131211yzaaxzaaxyaazayaxa)()()(322331132112233222211补充5、已知两个线性变换32133212311542322yyyxyyyxyyx,323323312211zzyzzyzzy321,zzz321,xxx求从到的 线性变换,AYX ,BZY ABZX 514232102A310102013B161109412316AB,1610941236321332
3、123211zzzxzzzxzzzx所以分析：6.设,)(10mmmaxaxaxf是阶方阵，An定义,)(10EaAaAaAfmmm, 35)(2xxxf3312A)(Af当时，求解：EAAAf35)(222 O7.设方阵A满足证明：AEA2,232OEAAA都可逆，分别表示出它们的为可逆矩阵。及并用证明：EAA232EAEA2)3(EAEA)3(21)3(211EAAEEAAEA4)2()2(,4222OEEAAAEEAEA4)(2(EEAEA)(21)2(1)2(EA8.利用初等行变换把下列矩阵化成行最简形矩阵：132126421321)1(A122rr 13rr 000000001321
4、03341431210110122413)2(B0334143121224130110112rr 02240422202111001101123rr 13rr 14rr 022404222021110011012) 1(r86200000002111001101232rr 244rr 00000431002111001101421r43rr 0000043100220104400132rr 31rr 9.对下列初等行变换，写出相应的初等方阵以及 和 之间的关系式。AB121121322101A122rr 12112330210113cc B1311233020011000100013E122r
5、r P10001200110000100001000014E13cc Q1000010000100101BPAQ 10.设,1APP其中2001,1141P求9A提示：1PPA111919)(PPPPPPPPA19PP11.设,200030004ABAAB2求B提示：ABEA)2(AEAB11)2(方法一：先求,)2(1 EAAEA1)2(再求方法二：直接求AEA1)2()2(AEA)2(1AEAE行41000100021)2(1EA210003000220003000441000100021B其中12.设,1210102312202321A利用初等行变换求1A)(EA 10001210010
6、01023001012200001232110001210010359400010122000012321133rr 0010122001035940100012100001232142rr 20101200410311001000121000012321234rr 243rr 102161000410311001000121000012321342rr 43rr 10216100061130100112160210204211032142rr 412rr 102161000611301001010001021120021322rr 313rr 212rr 102161000611301001
7、010001041120001)(1AE 102166113101041121A CCCD4321复习题一复习题一提示：提示：CBPBAP21,:3CPAP21111121211)(APPPAPC因为111212,PPPP所以1121APPCEEAEA0)2)( :4只有EEAA)3)(21(BBAABACnTTnTn1)()(:5)( :612nAAAEAEkkAAAAAAE212注意：这里的 就是 ，不是任意常数kkBAABAA6:71BEBA61EBEA6)(1EBEA)(6118.方法一：令,222112111XXXXOBAOsnsnEOOEEXXXXOBAO22211211snEBXOBXOAXEAX121122211121122121BXOXOXAX方法二：snsnsssnnnsnEOOBOEAOsnnnsnsnsnssOEAOEOOB12rr 11rB21rAsnnnsnsnsnssOAEOBOOE11OABOOBAO1119.利用第8题的结果0000000000000000121nnaaaaX?1X
