【正文】 120010100121 解 設(shè) ??????? 10 211A ??????? 30 122A ?????? ?? 12 131B ?????? ??? 30 322B 則 ???????????? 2121 BO BEAO EA ?????? ?? 22 2111 BAO BBAA 而 ?????? ???????? ????????? ????????? 42 2530 3212 1310 21211 BBA ?????? ????????? ????????? 90 3430 3230 1222 BA 所以 ???????????? 2121 BO BEAO EA ?????? ?? 22 2111 BAO BBAA ??????????????9000340042102521 即 ???????????????????????30003200121013013000120010100121??????????????903404102521 27? 取 ??????????? 10 01DCBA ? 驗(yàn)證 |||| |||| DC BADC BA ? 解 410 0120 0210100101002000021010010110100101????????DCBA 而 011 11|||| |||| ??DC BA 故 |||| |||| DC BADC BA ? 28? 設(shè)?????????? ??22023443OOA ? 求 |A8|及 A4 解 令 ?????? ?? 34 431A ??????? 22 022A 則 ??????? 21 AO OAA 故 8218 ??????? AO OAA ??????? 8281 AO OA 16828182818 10|||||||||| ??? AAAAA ??????????????????464444241422025005OOAOOAA 29? 設(shè) n 階矩陣 A 及 s 階矩陣 B 都可逆 ? 求 (1) 1??????? OB AO 解 設(shè) ??????????????43211 CC CCOB AO 則 ?????? OB AO ?????? 43 21 CC CC ?????????????? sn EO OEBCBC ACAC 21 43 由此得 ?????????snEBCOBCOACEAC2143???????????121413BCOCOCAC 所以 ???????????????? OA BOOB AO111 ? (2) 1??????? BC OA 解 設(shè) ??????????????43211 DD DDBC OA 則 ????????????? ??????????????? sn EO OEBDCDBDCD ADADDD DDBC OA 4231 2143 21 由此得 ???????????snEBDCDOBDCDOADEAD423121???????????????14113211BDCABDODAD 所以 ?????? ?????????????11111 BCAB OABC OA 30 求下列矩陣的逆陣 (1)??????????2500380000120025 解 設(shè) ??????? 12 25A ??????? 25 38B 則 ??????? ????????? ?? 52 2112 25 11A ??????? ????????? ?? 85 3225 38 11B 于是 ???????????????????????????????????????????850032000052002125003800001200251111BABA (2)??????????4121031200210001 解 設(shè) ??????? 21 01A ??????? 41 03B ??????? 21 12C 則 ?????? ?????????????????????????1111114121031200210001BCABOABCOA ??????????????????????411212458103161210021210001? 第三章 矩陣的初等變換與線性方程組 1 把下列矩陣化為行最簡形矩陣 (1) ??????????3403 13021201 解 ??????????3403 13021201 (下一步 r2?(?2)r1 r3?(?3)r1 ) ~ ???????????0200 31001201 (下一步 r2?(?1) r3?(?2) ) ~ ???????? ??0100 31001201 (下一步 r3?r2 ) ~ ???????? ??3000 31001201 (下一步 r3?3 ) ~ ???????? ??1000 31001201 (下一步 r2?3r3 ) ~ ???????? ?1000 01001201 (下一步 r1?(?2)r2 r1?r3 ) ~ ????????1000 01000001 (2) ????????????1740 34301320 解 ????????????1740 34301320 (下一步 ? r2?2?(?3)r1? r3?(?2)r1? ) ~ ???????????3100 31001320 (下一步 ? r3?r2? r1?3r2? ) ~ ????????0000 310010020 (下一步 ? r1?2? ) ~ ????????0000 31005010 (3)???????????????????12433023221453334311 解 ???????????????????12433023221453334311(下一步 ? r2?3r1? r3?2r1? r4?3r1? ) ~??????????????????1010500663008840034311(下一步 ? r2?(?4)? r3?(?3) ? r4?(?5)? ) ~???????????????22100221002210034311(下一步 ? r1?3r2? r3?r2? r4?r2? ) ~?????????? ? ??00000000002210032022 (4)????????????????34732038234202173132 解 ????????????????34732038234202173132(下一步 ? r1?2r2? r3?3r2? r4?2r2? ) ~???????????????1187701298804202111110(下一步 ? r2?2r1? r3?8r1? r4?7r1? ) ~?????????? ??41000410002020111110(下一步 ? r1?r2? r2?(?1)? r4?r3? ) ~?????????? ??? ?00000410001111020201(下一步 ? r2?r3? ) ~?????????? ? ?00000410003011020201 2 設(shè) ?????????????????????????987 654321100 010101100 001010 A 求 A 解 ????????100 001010 是初等矩陣 E(1 2) 其逆矩陣就是其本身 ????????100 010101 是初等矩陣 E(1 2(1)) 其逆矩陣是 E(1 2(?1)) ???????? ??100 010101 ???????? ??????????????????100 010101987 654321100 001010A ????????????????? ??????????287 221254100 010101987 321654 3 試?yán)镁仃嚨某醯茸儞Q 求下列方陣的逆矩陣 (1) ????????323 513123 解 ????????100 010001323 513123 ~??????????? 101 011001200 410123 ~ ????????? ???101200 2110102/102/3023 ~????????? ???2/102/110 2110102/922/7003 ~ ????????? ???2/102/1100 2110102/33/26/7001 故逆矩陣為????????????????21021211233267 (2)???????????????1210232112201023 解 ???????????????10000100001000011210232112201023 ~?????