2009-7-26
1
线性代数第 2讲作业的问题
2009-7-26
2
作业的问题作业中最大的问题就是,许多学生并没有将方程的增广矩阵,经过一系列行初等变换后,变化成行简化阶梯矩阵,
将任何一个矩阵经过一系列行初等变换,变化成行简化阶梯矩阵,是线性代数的基本技术,一定要掌握,
2009-7-26
3
行简化阶梯矩阵的例,
1 1 0 0 7 1
0 0 1 0 4 2
0 0 0 1 3 1
0 0 0 0 0 0
2009-7-26
4
不是行简化阶梯矩阵的例,
2 2 0 0 14 2
0 0 1 0 4 2
0 0 0 1 3 1
0 0 0 0 0 0
2009-7-26
5
不是行简化阶梯矩阵的例,
1 1 0 3 7 1
0 0 1 3 4 2
0 0 0 1 3 1
0 0 0 0 0 0
2009-7-26
6
不是行简化阶梯矩阵的例,
1 1 0 0 7 1
0 0 1 0 4 2
0 0 0 1 3 1
0 0 0 1 1 2
2009-7-26
7
不是行简化阶梯矩阵的例,
1 1 0 0 7 1
0 0 1 0 4 2
0 0 0 0 0 2
0 0 0 0 0 3
2009-7-26
8
习题 1
1 2 3
1 2 4
1 3 4
2 3 4
11
2 0,
22
11
2 3,
22
11
2 3,
22
11
2 0.
22
x x x
x x x
x x x
x x x
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11
2 0 0
22
11
2 0 3
22
11
0 2 3
22
11
0 2 0
22
1 1 1 1 6
11
2 0 3
22
11
0 2 3
22
11
0 2 0
22
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1 1 1 1 6
11
2 0 3
22
11
0 2 3
22
11
0 2 0
22
1 1 1 1 6
1 4 0 1 6
1 0 4 1 6
0 1 1 4 0
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11
1 1 1 1 6
1 4 0 1 6
1 0 4 1 6
0 1 1 4 0
1 1 1 1 6
0 5 1 0 12
0 1 5 0 12
0 1 1 4 0
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12
1 1 1 1 6
0 5 1 0 12
0 1 5 0 12
0 1 1 4 0
1 1 1 1 6
0 1 5 0 12
0 5 1 0 12
0 1 1 4 0
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1 1 1 1 6
0 1 5 0 12
0 5 1 0 12
0 1 1 4 0
1 0 4 1 6
0 1 5 0 12
0 0 24 0 48
0 0 4 4 12
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1 0 4 1 6
0 1 5 0 12
0 0 24 0 48
0 0 4 4 12
1 0 4 1 6
0 1 5 0 12
0 0 1 0 2
0 0 1 1 3
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1 0 4 1 6
0 1 5 0 12
0 0 1 0 2
0 0 1 1 3
1 0 0 1 2
0 1 0 0 2
0 0 1 0 2
0 0 0 1 1
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16
1 0 0 1 2
0 1 0 0 2
0 0 1 0 2
0 0 0 1 1
1 2 3 4
1 0 0 0 1
0 1 0 0 2
0 0 1 0 2
0 0 0 1 1
1,2,2,1x x x x
2009-7-26
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习题 2
1 2 3 4
2 3 4
1 2 4
2 3 4
2 3 4 4,
3,
3 3 1,
7 3 3,
x x x x
x x x
x x x
x x x
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18
1 2 3 4 4
0 1 1 1 3
1 3 0 3 1
0 7 3 1 3
1 2 3 4
2 3 4
1 2 4
2 3 4
2 3 4 4,
3,
3 3 1,
7 3 3,
x x x x
x x x
x x x
x x x
2009-7-26
19
1 2 3 4 4
0 1 1 1 3
1 3 0 3 1
0 7 3 1 3
1 2 3 4 4
0 1 1 1 3
0 5 3 1 3
0 7 3 1 3
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1 2 3 4 4
0 1 1 1 3
0 5 3 1 3
0 7 3 1 3
1 0 1 2 2
0 1 1 1 3
0 0 2 4 12
0 0 4 8 24
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1 0 1 2 2
0 1 1 1 3
0 0 2 4 12
0 0 4 8 24
1 0 1 2 2
0 1 1 1 3
0 0 1 2 6
0 0 1 2 6
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1 0 1 2 2
0 1 1 1 3
0 0 1 2 6
0 0 1 2 6
1 0 0 0 8
0 1 0 1 3
0 0 1 2 6
0 0 0 0 0
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.
,,26,3,8 4321
为任意常数k
kxkxkxx
1 0 0 0 8
0 1 0 1 3
0 0 1 2 6
0 0 0 0 0
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习题 3
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
2 3 5 3,
3 4 2 3 2,
2 8 8,
7 9 8 0,
x x x x
x x x x
x x x x
x x x x
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2 3 5 1 3
3 4 2 3 2
1 2 8 1 8
7 9 1 8 0
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
2 3 5 3,
3 4 2 3 2,
2 8 8,
7 9 8 0,
x x x x
x x x x
x x x x
x x x x
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2 3 5 1 3
3 4 2 3 2
1 2 8 1 8
7 9 1 8 0
1 2 8 1 8
3 4 2 3 2
2 3 5 1 3
7 9 1 8 0
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1 2 8 1 8
3 4 2 3 2
2 3 5 1 3
7 9 1 8 0
1 2 8 1 8
0 2 22 6 26
0 1 11 3 13
0 5 55 15 56
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1 2 8 1 8
0 2 22 6 26
0 1 11 3 13
0 5 55 15 56
1 2 8 1 8
0 1 11 3 13
0 1 11 3 13
0 1 11 3 56 / 5
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1 2 8 1 8
0 1 11 3 13
0 1 11 3 13
0 1 11 3 56 / 5
1 0 4 5 18
0 1 11 3 13
0 0 0 0 0
0 0 0 0 9 / 5
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1 0 4 5 18
0 1 11 3 13
0 0 0 0 0
0 0 0 0 9 / 5
1 0 4 5 18
0 1 11 3 13
0 0 0 0 1
0 0 0 0 0
方程无解
2009-7-26
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习题 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
10 11 11 0,
2 4 5 7 0,
3 3 3 2 0,
5 2 5 0.
x x x x
x x x x
x x x x
x x x x
2009-7-26
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1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
10 11 11 0,
2 4 5 7 0,
3 3 3 2 0,
5 2 5 0.
x x x x
x x x x
x x x x
x x x x
1 10 11 11 0
2 4 5 7 0
3 3 3 2 0
5 1 2 5 0
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1 10 11 11 0
2 4 5 7 0
3 3 3 2 0
5 1 2 5 0
1 10 11 11 0
0 24 27 29 0
0 27 30 31 0
0 51 57 60 0
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1 10 11 11 0
0 24 27 29 0
0 27 30 31 0
0 51 57 60 0
1 10 11 11 0
0 3 3 2 0
0 27 30 31 0
0 51 57 60 0
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1 10 11 11 0
0 3 3 2 0
0 27 30 31 0
0 51 57 60 0
1 10 11 11 0
0 1 1 2 / 3 0
0 27 30 31 0
0 51 57 60 0
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1 10 11 11 0
0 1 1 2 / 3 0
0 27 30 31 0
0 51 57 60 0
1 0 1 13 / 3 0
0 1 1 2 / 3 0
0 0 3 13 0
0 0 6 26 0
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1 0 1 13 / 3 0
0 1 1 2 / 3 0
0 0 3 13 0
0 0 6 26 0
1 0 1 1 3 / 3 0
0 1 1 2 / 3 0
0 0 1 1 3 / 3 0
0 0 0 0 0
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1 0 1 1 3 / 3 0
0 1 1 2 / 3 0
0 0 1 1 3 / 3 0
0 0 0 0 0
1 2 3 4
1 0 0 0 0
0 1 0 11 / 3 0
0 0 1 13 / 3 0
0 0 0 0 0
11 13
0,,,
33
x x k x k x k
k是任意常数
2009-7-26
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习题 5
1 2 3
1 2 3
2
1 2 3
1p x x x
x p x x p
x x p x p
2009-7-26
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2
1 1 1
11
11
p
B p p
pp
1 2 3
1 2 3
2
1 2 3
1p x x x
x p x x p
x x p x p
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2
1 1 1
11
11
p
B p p
pp
2
2
2 2 2 1
11
11
p p p p p
pp
pp
2009-7-26
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当 p2时,
方程无解
2
2
2 2 2 1
11
11
p p p p p
pp
pp
0 0 0 3
1 2 1 2
1 1 2 4
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当 p2时,
2
2
2 2 2 1
11
11
p p p p p
pp
pp
2
2
1 1 1 ( )
1
1 1,( )
2
11
up
pp
p p u p
p
pp
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2
2
1 1 1 ( )
1
1 1,( )
2
11
up
pp
p p u p
p
pp
2
1 1 1 ( )
0 1 0 ( )
0 0 1 ( )
up
p p u p
p p u p
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2
2
22
2
3 2 2 2
1
( ),
2
( 2) 1 1
()
2 2 2
( 2) 1
()
22
( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 )
2 2 2
pp
up
p
p p p p p
p u p
p p p
p p p p
p u p
pp
p p p p p p p
p p p
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2
1 1 1 ( )
0 1 0 ( )
0 0 1 ( )
up
p p u p
p p u p
2
2
1
1 1 1
2
1
0 1 0
2
( 1 ) ( 1 )
0 0 1
2
pp
p
p
p
p
pp
p
p
2009-7-26
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如 p=1,则
1111
0000
0000
B
方程有无穷多解,令 x2=k1,x3=k2,k1,k2为任意常数,则 x1=1?k1?k2,方程的解为
[1?k1?k2,k1,k2]
2009-7-26
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如 p?1且 p2则
2
2
1
1 1 1
2
1
0 1 0
2
( 1 ) ( 1 )
0 0 1
2
pp
p
p
p
p
pp
p
p
2
2
1
111
2
1
0 1 0
2
( 1 )
0 0 1
2
pp
p
p
p
p
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2
2
1
111
2
1
0 1 0
2
( 1 )
0 0 1
2
pp
p
p
p
p
2
1
1 0 0
2
1
0 1 0
2
( 1 )
0 0 1
2
p
p
p
p
p
2009-7-26
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由可得方程有唯一解
2
1
1 0 0
2
1
0 1 0
2
( 1 )
0 0 1
2
p
p
B
p
p
p
2
1 2 3
1 1 ( 1 )
,,
2 2 2
pp
x x x
p p p
2009-7-26
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习题 6
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3 6 2 1,
2 3 0,
5 10,
3 4 1.
x x x x
x x x x
x x x x q
x x px x
2009-7-26
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1 3 6 2 1
1 1 2 3 0
1 5 1 0 1
3 1 4 1
B
q
p
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3 6 2 1,
2 3 0,
5 10,
3 4 1.
x x x x
x x x x
x x x x q
x x px x
2009-7-26
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1 3 6 2 1
1 1 2 3 0
1 5 1 0 1
3 1 4 1
B
q
p
1 3 6 2 1
0 2 4 1 1
0 8 1 6 3 1
0 1 0 1 8 2 4
q
p
2009-7-26
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1 3 6 2 1
0 2 4 1 1
0 8 1 6 3 1
0 1 0 1 8 2 4
q
p
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 8 16 3 1
0 10 18 2 4
q
p
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1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 8 16 3 1
0 10 18 2 4
q
p
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 3
0 0 2 7 1
q
p
2009-7-26
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1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 3
0 0 2 7 1
q
p
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 2 7 1
0 0 0 7 3
p
q
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当 p?2时,无论 q取何值,方程都有唯一解,如
p=2,则这时如 q?2,方程无解
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 2 7 1
0 0 0 7 3
B
p
q
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 1
0 0 0 7 3
B
q
2009-7-26
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如 p=2,q=2,1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 1
0 0 0 0 0
B
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 1 1 / 7
0 0 0 0 0
B
2009-7-26
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如 p=2,q=2,
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 1 1 / 7
0 0 0 0 0
B
1 0 0 0 0
0 1 2 0 6 / 14
0 0 0 1 1 / 7
0 0 0 0 0
B
2009-7-26
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方程有无穷多组解,x3为自由变元,令 x3=k,k为任意常数,则
1 2 3 4
31
0,2,,
77
x x k x k x
1 0 0 0 0
0 1 2 0 6 / 14
0 0 0 1 1 / 7
0 0 0 0 0
B
2009-7-26
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如 p?2,方程有唯一解,这时
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 2 7 1
0 0 0 7 3
B
p
q
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 1 7 /( 2 ) 1 /( 2 )
0 0 0 1 ( 3 ) / 7
pp
q
2009-7-26
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1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 1 7 /( 2 ) 1 /( 2 )
0 0 0 1 ( 3 ) / 7
pp
q
321
2 2 2
3211
2 2 7 2
321
2 2 2
3
7
2
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
qq
qq
p
qq
p p p
q
2009-7-26
63
今天作业,重做 1,2,3,4题
5,6题
1
线性代数第 2讲作业的问题
2009-7-26
2
作业的问题作业中最大的问题就是,许多学生并没有将方程的增广矩阵,经过一系列行初等变换后,变化成行简化阶梯矩阵,
将任何一个矩阵经过一系列行初等变换,变化成行简化阶梯矩阵,是线性代数的基本技术,一定要掌握,
2009-7-26
3
行简化阶梯矩阵的例,
1 1 0 0 7 1
0 0 1 0 4 2
0 0 0 1 3 1
0 0 0 0 0 0
2009-7-26
4
不是行简化阶梯矩阵的例,
2 2 0 0 14 2
0 0 1 0 4 2
0 0 0 1 3 1
0 0 0 0 0 0
2009-7-26
5
不是行简化阶梯矩阵的例,
1 1 0 3 7 1
0 0 1 3 4 2
0 0 0 1 3 1
0 0 0 0 0 0
2009-7-26
6
不是行简化阶梯矩阵的例,
1 1 0 0 7 1
0 0 1 0 4 2
0 0 0 1 3 1
0 0 0 1 1 2
2009-7-26
7
不是行简化阶梯矩阵的例,
1 1 0 0 7 1
0 0 1 0 4 2
0 0 0 0 0 2
0 0 0 0 0 3
2009-7-26
8
习题 1
1 2 3
1 2 4
1 3 4
2 3 4
11
2 0,
22
11
2 3,
22
11
2 3,
22
11
2 0.
22
x x x
x x x
x x x
x x x
2009-7-26
9
11
2 0 0
22
11
2 0 3
22
11
0 2 3
22
11
0 2 0
22
1 1 1 1 6
11
2 0 3
22
11
0 2 3
22
11
0 2 0
22
2009-7-26
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1 1 1 1 6
11
2 0 3
22
11
0 2 3
22
11
0 2 0
22
1 1 1 1 6
1 4 0 1 6
1 0 4 1 6
0 1 1 4 0
2009-7-26
11
1 1 1 1 6
1 4 0 1 6
1 0 4 1 6
0 1 1 4 0
1 1 1 1 6
0 5 1 0 12
0 1 5 0 12
0 1 1 4 0
2009-7-26
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1 1 1 1 6
0 5 1 0 12
0 1 5 0 12
0 1 1 4 0
1 1 1 1 6
0 1 5 0 12
0 5 1 0 12
0 1 1 4 0
2009-7-26
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1 1 1 1 6
0 1 5 0 12
0 5 1 0 12
0 1 1 4 0
1 0 4 1 6
0 1 5 0 12
0 0 24 0 48
0 0 4 4 12
2009-7-26
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1 0 4 1 6
0 1 5 0 12
0 0 24 0 48
0 0 4 4 12
1 0 4 1 6
0 1 5 0 12
0 0 1 0 2
0 0 1 1 3
2009-7-26
15
1 0 4 1 6
0 1 5 0 12
0 0 1 0 2
0 0 1 1 3
1 0 0 1 2
0 1 0 0 2
0 0 1 0 2
0 0 0 1 1
2009-7-26
16
1 0 0 1 2
0 1 0 0 2
0 0 1 0 2
0 0 0 1 1
1 2 3 4
1 0 0 0 1
0 1 0 0 2
0 0 1 0 2
0 0 0 1 1
1,2,2,1x x x x
2009-7-26
17
习题 2
1 2 3 4
2 3 4
1 2 4
2 3 4
2 3 4 4,
3,
3 3 1,
7 3 3,
x x x x
x x x
x x x
x x x
2009-7-26
18
1 2 3 4 4
0 1 1 1 3
1 3 0 3 1
0 7 3 1 3
1 2 3 4
2 3 4
1 2 4
2 3 4
2 3 4 4,
3,
3 3 1,
7 3 3,
x x x x
x x x
x x x
x x x
2009-7-26
19
1 2 3 4 4
0 1 1 1 3
1 3 0 3 1
0 7 3 1 3
1 2 3 4 4
0 1 1 1 3
0 5 3 1 3
0 7 3 1 3
2009-7-26
20
1 2 3 4 4
0 1 1 1 3
0 5 3 1 3
0 7 3 1 3
1 0 1 2 2
0 1 1 1 3
0 0 2 4 12
0 0 4 8 24
2009-7-26
21
1 0 1 2 2
0 1 1 1 3
0 0 2 4 12
0 0 4 8 24
1 0 1 2 2
0 1 1 1 3
0 0 1 2 6
0 0 1 2 6
2009-7-26
22
1 0 1 2 2
0 1 1 1 3
0 0 1 2 6
0 0 1 2 6
1 0 0 0 8
0 1 0 1 3
0 0 1 2 6
0 0 0 0 0
2009-7-26
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.
,,26,3,8 4321
为任意常数k
kxkxkxx
1 0 0 0 8
0 1 0 1 3
0 0 1 2 6
0 0 0 0 0
2009-7-26
24
习题 3
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
2 3 5 3,
3 4 2 3 2,
2 8 8,
7 9 8 0,
x x x x
x x x x
x x x x
x x x x
2009-7-26
25
2 3 5 1 3
3 4 2 3 2
1 2 8 1 8
7 9 1 8 0
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
2 3 5 3,
3 4 2 3 2,
2 8 8,
7 9 8 0,
x x x x
x x x x
x x x x
x x x x
2009-7-26
26
2 3 5 1 3
3 4 2 3 2
1 2 8 1 8
7 9 1 8 0
1 2 8 1 8
3 4 2 3 2
2 3 5 1 3
7 9 1 8 0
2009-7-26
27
1 2 8 1 8
3 4 2 3 2
2 3 5 1 3
7 9 1 8 0
1 2 8 1 8
0 2 22 6 26
0 1 11 3 13
0 5 55 15 56
2009-7-26
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1 2 8 1 8
0 2 22 6 26
0 1 11 3 13
0 5 55 15 56
1 2 8 1 8
0 1 11 3 13
0 1 11 3 13
0 1 11 3 56 / 5
2009-7-26
29
1 2 8 1 8
0 1 11 3 13
0 1 11 3 13
0 1 11 3 56 / 5
1 0 4 5 18
0 1 11 3 13
0 0 0 0 0
0 0 0 0 9 / 5
2009-7-26
30
1 0 4 5 18
0 1 11 3 13
0 0 0 0 0
0 0 0 0 9 / 5
1 0 4 5 18
0 1 11 3 13
0 0 0 0 1
0 0 0 0 0
方程无解
2009-7-26
31
习题 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
10 11 11 0,
2 4 5 7 0,
3 3 3 2 0,
5 2 5 0.
x x x x
x x x x
x x x x
x x x x
2009-7-26
32
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
10 11 11 0,
2 4 5 7 0,
3 3 3 2 0,
5 2 5 0.
x x x x
x x x x
x x x x
x x x x
1 10 11 11 0
2 4 5 7 0
3 3 3 2 0
5 1 2 5 0
2009-7-26
33
1 10 11 11 0
2 4 5 7 0
3 3 3 2 0
5 1 2 5 0
1 10 11 11 0
0 24 27 29 0
0 27 30 31 0
0 51 57 60 0
2009-7-26
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1 10 11 11 0
0 24 27 29 0
0 27 30 31 0
0 51 57 60 0
1 10 11 11 0
0 3 3 2 0
0 27 30 31 0
0 51 57 60 0
2009-7-26
35
1 10 11 11 0
0 3 3 2 0
0 27 30 31 0
0 51 57 60 0
1 10 11 11 0
0 1 1 2 / 3 0
0 27 30 31 0
0 51 57 60 0
2009-7-26
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1 10 11 11 0
0 1 1 2 / 3 0
0 27 30 31 0
0 51 57 60 0
1 0 1 13 / 3 0
0 1 1 2 / 3 0
0 0 3 13 0
0 0 6 26 0
2009-7-26
37
1 0 1 13 / 3 0
0 1 1 2 / 3 0
0 0 3 13 0
0 0 6 26 0
1 0 1 1 3 / 3 0
0 1 1 2 / 3 0
0 0 1 1 3 / 3 0
0 0 0 0 0
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1 0 1 1 3 / 3 0
0 1 1 2 / 3 0
0 0 1 1 3 / 3 0
0 0 0 0 0
1 2 3 4
1 0 0 0 0
0 1 0 11 / 3 0
0 0 1 13 / 3 0
0 0 0 0 0
11 13
0,,,
33
x x k x k x k
k是任意常数
2009-7-26
39
习题 5
1 2 3
1 2 3
2
1 2 3
1p x x x
x p x x p
x x p x p
2009-7-26
40
2
1 1 1
11
11
p
B p p
pp
1 2 3
1 2 3
2
1 2 3
1p x x x
x p x x p
x x p x p
2009-7-26
41
2
1 1 1
11
11
p
B p p
pp
2
2
2 2 2 1
11
11
p p p p p
pp
pp
2009-7-26
42
当 p2时,
方程无解
2
2
2 2 2 1
11
11
p p p p p
pp
pp
0 0 0 3
1 2 1 2
1 1 2 4
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43
当 p2时,
2
2
2 2 2 1
11
11
p p p p p
pp
pp
2
2
1 1 1 ( )
1
1 1,( )
2
11
up
pp
p p u p
p
pp
2009-7-26
44
2
2
1 1 1 ( )
1
1 1,( )
2
11
up
pp
p p u p
p
pp
2
1 1 1 ( )
0 1 0 ( )
0 0 1 ( )
up
p p u p
p p u p
2009-7-26
45
2
2
22
2
3 2 2 2
1
( ),
2
( 2) 1 1
()
2 2 2
( 2) 1
()
22
( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 )
2 2 2
pp
up
p
p p p p p
p u p
p p p
p p p p
p u p
pp
p p p p p p p
p p p
2009-7-26
46
2
1 1 1 ( )
0 1 0 ( )
0 0 1 ( )
up
p p u p
p p u p
2
2
1
1 1 1
2
1
0 1 0
2
( 1 ) ( 1 )
0 0 1
2
pp
p
p
p
p
pp
p
p
2009-7-26
47
如 p=1,则
1111
0000
0000
B
方程有无穷多解,令 x2=k1,x3=k2,k1,k2为任意常数,则 x1=1?k1?k2,方程的解为
[1?k1?k2,k1,k2]
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48
如 p?1且 p2则
2
2
1
1 1 1
2
1
0 1 0
2
( 1 ) ( 1 )
0 0 1
2
pp
p
p
p
p
pp
p
p
2
2
1
111
2
1
0 1 0
2
( 1 )
0 0 1
2
pp
p
p
p
p
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49
2
2
1
111
2
1
0 1 0
2
( 1 )
0 0 1
2
pp
p
p
p
p
2
1
1 0 0
2
1
0 1 0
2
( 1 )
0 0 1
2
p
p
p
p
p
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50
由可得方程有唯一解
2
1
1 0 0
2
1
0 1 0
2
( 1 )
0 0 1
2
p
p
B
p
p
p
2
1 2 3
1 1 ( 1 )
,,
2 2 2
pp
x x x
p p p
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51
习题 6
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3 6 2 1,
2 3 0,
5 10,
3 4 1.
x x x x
x x x x
x x x x q
x x px x
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52
1 3 6 2 1
1 1 2 3 0
1 5 1 0 1
3 1 4 1
B
q
p
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
3 6 2 1,
2 3 0,
5 10,
3 4 1.
x x x x
x x x x
x x x x q
x x px x
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53
1 3 6 2 1
1 1 2 3 0
1 5 1 0 1
3 1 4 1
B
q
p
1 3 6 2 1
0 2 4 1 1
0 8 1 6 3 1
0 1 0 1 8 2 4
q
p
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54
1 3 6 2 1
0 2 4 1 1
0 8 1 6 3 1
0 1 0 1 8 2 4
q
p
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 8 16 3 1
0 10 18 2 4
q
p
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55
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 8 16 3 1
0 10 18 2 4
q
p
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 3
0 0 2 7 1
q
p
2009-7-26
56
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 3
0 0 2 7 1
q
p
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 2 7 1
0 0 0 7 3
p
q
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57
当 p?2时,无论 q取何值,方程都有唯一解,如
p=2,则这时如 q?2,方程无解
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 2 7 1
0 0 0 7 3
B
p
q
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 1
0 0 0 7 3
B
q
2009-7-26
58
如 p=2,q=2,1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 7 1
0 0 0 0 0
B
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 1 1 / 7
0 0 0 0 0
B
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59
如 p=2,q=2,
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 0 1 1 / 7
0 0 0 0 0
B
1 0 0 0 0
0 1 2 0 6 / 14
0 0 0 1 1 / 7
0 0 0 0 0
B
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60
方程有无穷多组解,x3为自由变元,令 x3=k,k为任意常数,则
1 2 3 4
31
0,2,,
77
x x k x k x
1 0 0 0 0
0 1 2 0 6 / 14
0 0 0 1 1 / 7
0 0 0 0 0
B
2009-7-26
61
如 p?2,方程有唯一解,这时
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 2 7 1
0 0 0 7 3
B
p
q
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 1 7 /( 2 ) 1 /( 2 )
0 0 0 1 ( 3 ) / 7
pp
q
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62
1 0 0 7 / 2 1 / 2
0 1 2 1 / 2 1 / 2
0 0 1 7 /( 2 ) 1 /( 2 )
0 0 0 1 ( 3 ) / 7
pp
q
321
2 2 2
3211
2 2 7 2
321
2 2 2
3
7
2
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
p
p p p
q
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63
今天作业,重做 1,2,3,4题
5,6题
