?218?2?30 (3)?3?25?103?37?7?5?80?20??

?218?2?30 解 ?3?25?103?37?7?5?(下一步80?20?? r12r4 r22r4 r33r4 )

?012?1?0?3?63 ~?0?2?42?1032??0?0 ~?0?1??0?0 ~?0?1?10001000200320037??5?(下一步0?0?? r23r1 r32r1 )

?17?016?(下一步014?20???10027?1? 0?0?? r216r4 r316r2 )

?1?0 ~?0?0?010032002?1000?7?1?0??

07?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

可使

?1?23k? 11 设A???12k?3??k?23??? (1)R(A)1 (2)R(A)

问k为何值

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 (2)当k (3)当k 12

求解下列齐次线性方程组:

1时

R(A)1

1时2时

R(A) R(A)

2 3

2且k1且k??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 A???211121??11??~??0101?31?0??2212????001?4/31?????x1?4x于是 ??x2?34?3

?x4x4x?3??x4?34x4故方程组的解为

?x?4?1??3? ??xx2???k???43??(k为任意常数) ?3?x?4???3??1?? (2)??x1?2x2?x3?x4?0?3x1?6x2?x3?3x4?0

??5x1?10x2?x3?5x4?0 解 对系数矩阵A进行初等行变换 A??121?1???36?1?3?~?0120?1??5101?5????000010?0???于是 ?x1??2x?xx2?x42?2

?x?0?x34?x4 有

有

故方程组的解为

?x1???2??1??x??1??0?2 ???k1???k2??(k1

k2为任意常数)

?x?x3??04??0?0???1???2x1?3x2?x3?5x4 (3)??0?3x?x?2x4x123?7x4?0

?1?x2?3x3?6x4?0?x1?2x2?4x3?7x4?0 解 对系数矩阵A进行初等行变换??0 A?2?331?213?57???110?410000???1?24?6?~??07?0???00100?1???于是 ?x?x1?02?0

?x?0?x34?0故方程组的解为 ?x1 ??x?02?0

?x?0?x34?0?3x1 (4)??4x2?5x3?7x4?0?2x?3x2?3x3?2x4?041

?x1?11x2?13x3?16x4?0?7x1?2x2?x3?3x4?0有