内容发布更新时间 : 2024/5/28 11:24:36星期一 下面是文章的全部内容请认真阅读。
?012?1?0?3?63 ~?0?2?42?1032??0?0 ~?0?1??0?0 ~?0?1?10001000200320037??5?(下一步? r?3r? r?2r? )
2131
0?0???17?016?(下一步? r?16r? r?16r? )
2432
014?20???10027?1? 0?0??0?7?? 1?0???1?0 ~?0?0?010032002?10007?5矩阵的秩为3? 580?70?0是一个最高阶非零子式?
320 10? 设A、B都是m?n矩阵? 证明A~B的充分必要条件是R(A)?R(B)?
证明 根据定理3? 必要性是成立的?
充分性? 设R(A)?R(B)? 则A与B的标准形是相同的? 设A与B的标准形为D? 则有
A~D? D~B?
由等价关系的传递性? 有A~B?
?1?23k? 11? 设A???12k?3?? 问k为何值? 可使
?k?23??? (1)R(A)?1? (2)R(A)?2? (3)R(A)?3?
k?1?23k?r?1?1??? 解 A???12k?3?~ ?0k?1k?1?k?23??00?(k?1)(k?2)????? (1)当k?1时? R(A)?1? (2)当k??2且k?1时? R(A)?2? (3)当k?1且k??2时? R(A)?3?
12? 求解下列齐次线性方程组:
??x1?x2?2x3?x4?0 (1)?2x1?x2?x3?x4?0?
??2x1?2x2?x3?2x4?0 解 对系数矩阵A进行初等行变换? 有
0??112?1??10?1 A??211?1?~?013?1??
?2212??001?4/3??????x?4x?134?x??3x4于是 ?2?
4?x3?x4?x?3?4x4故方程组的解为
?4??x1??3??x???3? ?2??k?4?(k为任意常数)?
?x3????x4??3??1???x1?2x2?x3?x4?0 (2)?3x1?6x2?x3?3x4?0?
??5x1?10x2?x3?5x4?0 解 对系数矩阵A进行初等行变换? 有
?121?1??120?1? A??36?1?3?~?0010??
?5101?5??0000??????x1??2x2?x4?x?x于是 ?22?
x3?0?x?x?44故方程组的解为
?x1???2??1??x??1??0? ?2??k1???k2??(k1? k2为任意常数)?
00?x3??0??1??x4??????2x1?3x2?x3?5x4?0?3x?x?2x3?7x4?0 (3)?12?
4x1?x2?3x3?6x4?0?x?2x?4x?7x?0?1234 解 对系数矩阵A进行初等行变换? 有 ?23?31 A??41?1?2??15??12?7?~?0?36??0?04?7???010000100?0?? 0?1??
?x1?0?x2?0于是 ??
x3?0?x?0?4故方程组的解为
?x1?0?x?0 ?2?
x?0?x3?0?4?3x1?4x2?5x3?7x4?0?2x?3x2?3x3?2x4?0 (4)?1?
4x?11x?13x3?16x4?0?7x1?2x2?x?3x4?0?123 解 对系数矩阵A进行初等行变换? 有 ?34?5?2?33 A??411?13?7?21??17???2?~?016???3???0?00?3171?1917000013?17?20?? ?17?00??
?x?3x?13x?1173174?19x?20xx?于是 ?2?
173174?x?x?x3?x3?44故方程组的解为
?3???13??x1??17??17??x??19??20? ?2??k1???k2???(k1? k2为任意常数)?
?x3??17??17??x4??1??0??0??1?
13? 求解下列非齐次线性方程组:
??4x1?2x2?x3?2 (1)?3x1?1x2?2x3?10?
??11x1?3x2?8 解 对增广矩阵B进行初等行变换? 有
?42?12??13?3?8? B??3?1210?~?0?101134??
?11308?000?6????于是R(A)?2? 而R(B)?3? 故方程组无解?
?2x?3y?z?4?x?2y?4z??5 (2)??
3x?8y?2z?13?4x?y?9z??6? 解 对增广矩阵B进行初等行变换? 有 ?231?1?24 B??38?2?4?19?4??1?5?~?013??0??6???001002?100?1?2?? 0?0????x??2z?1于是 ?y?z?2?
??z?z?x???2???1?即 ?y??k?1???2?(k为任意常数)?
?z??1??0?????????2x?y?z?w?1 (3)?4x?2y?2z?w?2?
??2x?y?z?w?1 解 对增广矩阵B进行初等行变换? 有