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2015《线性代数》清考模拟试卷 (1)

2015《线性代数》清考模拟试卷 (1)
2015《线性代数》清考模拟试卷 (1)

2015线性代数清考模拟试卷

一.选择题

1.设A 为n 阶矩阵,且2=A ,则AA '=( ). (A )4 (B )8 (C )16 (D )32 2.当向量组 12m ααα,,,线性相关时,使等式11220m m k k k ααα+++=成立的常数

12m k k k ,,,是( )

. (A )任意一组常数; (B )任意一组不全为零的常数; (C )某些特定的不全为零的常数; (D )唯一的一组不全为零的常数. 3.设A ,B 均为n 阶方阵,下列各形式总是成立的是( ). (A )A B A B +=+ (B )()AB A B '''= (C )2

2

2

()2A B A AB B +=++ (D )AB BA = 4.当向量组 12m ααα,,,线性相关时,使等式11220m m k k k ααα+++=成立的常数

12m k k k ,,,是( )

. (A )任意一组常数; (B )任意一组不全为零的常数; (C )某些特定的不全为零的常数; (D )唯一的一组不全为零的常数.

5.已知非齐次线性方程组AX b =的两个解1212,()X X X X ≠,则( )一定是

AX b =的解.

(A )12X X + (B )12X X - (C )1223X X - (D )12()2X X +

6.设2λ=是非零方阵A 的一个特征值,则矩阵1

A -有一个特征值等于( ).

()1A 1()

2B 1

()3C 1

()4

D

二.填空题

1.排列6427531的逆序数,即(6427531)τ= .

2.如果齐次线性方程组12312312

3000

x x x x x x x x x λλλ++=??

++=??++=?有非零解,则λ= .

3.若矩阵121211x y ????

=

? ?---????

,则x = ,y = .

4.若向量组121,24k αα????

== ? ?-????

线性相关,则k = .

5.若n 阶实对称矩阵A 满足2

1A =,则A 的特征值为 .

6.二次型22

1212

(,)2f x x x x =+所对应的对称矩阵为 . 三.计算题

1.计算行列式

41241202105200117

D =

.

2.设设????

? ??=110021001A ,用初等变换法求1

A -.

解:

3.设1211A ??= ?-??,32a b B ??

= ???

,若矩阵AB BA =,求a ,b 的值.

解:

四.计算题

1.已知矩阵112210

2151203131

1

041A ??

?

-

?

= ?- ?

-??

,求()R A 及其列向量组的一个极大无关组. 解:

2.设???

?

??-=3111A ,求A 的特征值和特征向量. 解:

3.解非齐次线性方程组123123

12312323424538213496

x x x x x x x x x x x x ++=??-+=-??+-=??-+=-?的通解.

解:

五.证明题

设向量组123,,ααα线性无关,且112223331,,βααβααβαα=+=+=+,证明:向量组123,,βββ也线性无关.

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