?23??12?3?2??1?3?1?3?7??12??0?2?4??23r?r12?830?3?2???1?3743???0?2?4??12?r2?2r1?1?3?7??0?1r?3r31?830?0?8?r4?r1??0?5743???0?2?4??111? ?212?014??004??00?2?
20?2??014??0?2?4??111?8912??767???12?r3?8r2?0?1r4?5r2?00??00??12r3?r4??0?1r2?r4?00?r1?2r4??00?10??0?1r1?2r2?00??00?0?2?4??12??111??0?1r?r34?014?00???00212???4??12??10?3?1?0?1r?r3?220?2?200????00014??000??1??r2?00?2??0120?2?r3?0?2??014???0000000??1??1002??0c?2c?c?4c5234?010?1?0???00014???0000??1000? ?0100?0010??三、用矩阵的初等变换，求矩阵的逆矩阵
?3?20?1???0221??A?? ?1?2?3?2???01?21???3?20?11?210?02?1?2?3?20??01210?000??1?2?3?20??100?210?02r?r13?010?3?20?11???01001?210??000110010??100? ?000?001??0??1? ?10?30?0100??000010?1?2?3?2?21?02r3?3r1?0495??0121?0??1?2?3?2??0?21?01r?r24?10?30?0495???020001?21???1?2?3?2?r3?4r2?012111r4?2r2?00??00?2?1??1?2?3r1?2r4?2?01r2?r4?001040??1?2?3?2??1?21?01r?2r43?10?3?4?0011???00010?2?01??2000000100??1? ?10?3?4?21?6?10??1?1?2000010?11?20??1?2??0?2?1611?r1?3r3?010?1?136?r2?2r3?00?2??00010?1?10?1?136?r3?r4???000121?6?10??????0??100011?2?4?r?2r?0100010?1??12??0010?1?136?
??000121?6?10?????11?2?4?A?1??010?1?????1?136? ???21?6?10????11?1??022??101?X??110?，求X
??10?????1???014??00121?6?10????四、已知?11?1101??11?1101??11?1101??022110?r?r?022110?r?r?022110?31?32?uuuuruuuur??u?u??1?10014??0?21?113??003023???????2??11012????3?11?1101????r2?2r3?1?1r3??022110?0201??2???r?r333uuuuur?1uuuuuuur?2??21??0010?1?3???0010?3??215????110121003????326????1?1111?r2?010??1?r?rr10??1?120uuuuu???22626uuuuur?????22 ??1?1??0010??0010?33?????1?2?1故X???2??0?5?3?6?1??1?
?6?21?3?线性代数练习题 第二章 矩 阵
系 专业 班 姓名 学号 第三节（二） 矩 阵 的 秩
一．选择题
1．设A，B都是n阶非零矩阵，且AB = 0，则A和B的秩 [ D ] （A）必有一个等于零 （B）都等于n （C）一个小于n，一个等于n （D）都不等于n 2．设m?n矩阵A的秩为s ，则 [ C ] （A）A的所有s－1阶子式不为零 （B）A的所有s阶子式不为零 （C）A的所有s +1阶子式为零 （D）对A施行初等行变换变成???Es?00?? ?0??11213???3．欲使矩阵?2s126?的秩为2，则s，t满足 [ C ]
?455t12???（A）s = 3或t = 4 （B）s = 2或t = 4 （C）s = 3且t = 4 （D）s = 2且t = 4 4．设A是m?n矩阵，B是n?m矩阵，则 [ B ] （A）当m?n时，必有行列式|AB|?0 （B）当m?n时，必有行列式|AB|?0 （C）当n?m时，必有行列式|AB|?0 （D）当n?m时，必有行列式|AB|?0
?aa12a13?a22a23??015．设A??11?aa?2122a???a2123?，B??aa12a??0??13?，P1??100?，?a31a32a33???11?a31?a11a32?a12a33?a13????001???100?P??010?2??，则必有B? [ C ]??101??（A）AP1P2 （B）AP2P1 （C）P1P2A （D）P2P1A 二．填空题：
?1．设A??3102??1?12?1??，则R(A)? 2
??13?44????121?2．已知A??23a?2????1a?2?的秩为2，则a 应满足 a=-1或3
??2a?2?1???三、计算题：
??21837?1． 设A??2?307?5????3?2580?，求R(A)。
??10320???
?218??2?30?3?25??103?37??103??7?5??2?30r?r14?80?3?25???21820???20?320??10?r2?2r1??7?5??0?3?63?5?r?3r 31???800?2?420???r?2r41??37?2?17??01?03212?10000000??1??7?8?0r?r2?414?70????016??03212?10000000??7? 14??0??320??10?1???2?17?r3?2r2?0?01r2?r4?0?2?420?r4?3r2?0???0?3?63?5???0???故R(A)=3
?1?23k???2．设A ???12k?3?，问k为何值，可使 ⑴ R(A)?1 ⑵R(A)?2 ⑶R(A)?3
?k?23????23k??23k?1?23k??1?1???r2?r1?????12k?302k?23k?3r?r02k?23k?3????32??
1?2??k?23?r3?kr?00?3(k?2)(k?1)????02k?23?3k???(1) R(A)=1当且仅当
2k?2?0??k?1 ??3(k?2)(k?1)?0?(2)由(1)可知R(A)=2当且仅当k=-2 (3)R(A)=3当且仅当k?1且k??2
线性代数练习题 第二章 矩 阵
系 专业 班 姓名 学号
第四节 矩阵的分块
?P?10??P0??1若A?? ?，则A???1?0Q0Q?????0若A???QP??0Q?1??1? ?，则A???10?0??P
线性代数第二章矩阵
