【正文】 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 ~?????。 解 根據(jù)第 6 題結(jié)果 有 nnnnnnnnnnaaanaaanaaaD)( )1()( )1( 11 11)1(1112)1(1????????????????????????????????????? 此行列式為范德蒙德行列式 ??????? ??????? 112)1(1 )]1()1[()1( jinnnn jaiaD ?????? ????112 )1( )]([)1(jinnn ji ????????????? ??????112 1 )1(2 )1( )()1()1(jinnnnn ji ????? ?? 11 )(jin ji? (4)nnnnndcdcbabaD????????????? 11112。 證明 444422221111dcba dcbadcba )()()(0)()()(001111222222222 addaccabbaddaccabb adacab?????? ???? )()()(111))()((222 addaccabb dcbadacab ??????? ))(())((00111))()((abdbddabcbcc bdbcadacab ?????? ?????? )()( 11))()()()(( abddabccbdbcadacab ?????????? =(a?b)(a?c)(a?d)(b?c)(b?d)(c?d)(a?b?c?d)? (5)1221 1 000 00 10 00 01axaaaa xxxnnn ???????? ??????????????????? ????????xn?a1xn?1? ? ? ? ?an?1x?an ? 證明 用數(shù)學(xué)歸納法證明 當(dāng) n?2 時 ? 212122 1 axaxaxaxD ??????? 命題成立 ? 假 設(shè)對于 (n?1)階行列式命題成立 ? 即 Dn?1?xn?1?a1 xn?2? ? ? ? ?an?2x?an?1? 則 Dn按第一列展開 有 1 11 00 1 00 01)1( 11 ???? ??????????????????? ??????? ?? xxaxDD nnnn ?xD n?1?an?xn?a1xn?1? ? ? ? ?an?1x?an ? 因此 ? 對于 n 階行列式命題成立 ? 6? 設(shè) n 階行列式 D?det(aij), 把 D 上下翻轉(zhuǎn)、或逆時針旋轉(zhuǎn)90?、或依副對角線翻轉(zhuǎn) ? 依次得 nnnnaaaaD11111 ??? ????????????? ? 11112 nnnnaaaaD??? ????????????? ? 11113 aaaaDnnnn??? ????????????? ? 證明 DDD nn 2 )1(21 )1( ???? ? D3?D ? 證明 因為 D?det(aij)? 所以 nnnnnnnnnnaaaaaaaaaaD2211111111111 )1( ??? ???????????????????? ????????????? ? ???????????????????????????? ?? )1()1(331