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厦门大学网络教育第一学期考试真题 线性代数

厦门大学网络教育第一学期考试真题 线性代数
厦门大学网络教育第一学期考试真题 线性代数

1.下列排列中,()是四级奇排列。

A4321

2.若(-1)。。。是五阶行列式【。。。】的一项,则k,l之值及该项符号为()

B k=2,l=3,符号为负

3.行列式【k-12。。。】的充分必要条件是()

C k不等于-1且k不等于3

4.若行列式D=【a11a12a13。。。】=M不等于0,则D1=【2a112a122a13。。。】=()C8M

5.行列式【0111】

1011

1101

1110=()

D-3

6.当a=()时,行列式【-1a2…】=0

B1

7.如果行列式【a11a12a13…】=d则【3a313a323a33…】=()

B6d

8.当a=()时,行列式【a11…】=0

A1

9.行列式【12564278。。。】的值为()

A12

10.行列式【a00b…】中g元素的代数余子式为()

B bde-bcf

11.设f(x)=【112。。。】则f(x)=0的根为()

C1,-1,2,-2

12.行列式【0a10…0。。。】=()

D(-1)n+1a1a2…an-1an1

13.行列式【a0b0…】=()

D(ad-bc)(xv-yu)

14.~不能取()时,方程组~X1+X2+X3=0…只有0解

B2

15.若三阶行列式D的第三行的元素依次为1,2,3它们的余子式分别为2,3,4,则D=() B8

16.设行列式【a11a12a13…】=1,则【2a113a11-4a12a13…】=()

D-8

1.线性方程组x1+x2=1…解的情况是()

A无解

2.若线性方程组AX=B的增广矩阵A经初等行变换化为A-【1234…】,当~不等于()时,

此线性方程组有唯一解

B0,1

3.已知n元线性方程组AX=B,其增广矩阵为A,当()时,线性方程组有解。

C r(A)=r(A)

4.设A为m*n矩阵,则齐次线性方程组AX=0仅有零解的充分条件是()

A A的列向量线性无关

5.非齐次线性方程组AX=B中,A和增广矩阵A的秩都是4,A是4*6矩阵,则下列叙述

正确的是()

B方程组有无穷多组解

6.设线性方程组AX=B有唯一解,则相应的齐次方程AX=0()

C只有零解

7.线性方程组AX=0只有零解,则AX=B(B不等于0)

B可能无解

8.设有向量组a1,a2,a3和向量B

A1=(1,1,1)a2=(1,1,0)a3=(1,0,0)B=(0,3,1)

则向量B由向量a1,a2,a3的线性表示是()

A B=a1+2a2-3a3

9.向量组a1=(1.1.1)(0.2.5)(1.3.6)是()

A线性相关

10.下列向量组线性相关的是()

C(7.4.1),(-2.1.2),(3.6.5)

11.向量组a1.a2…ar线性无关的充要条件是()

B向量线的秩等于它所含向量的个数

12.向量组B1.B2…Bt可由a1.a2…as线性表示出,且B1.B2…Bt线性无关,则s与t的关系

为()

D s≥t

13.n个向量a1.a2…an线性无关,去掉一个向量an,则剩下的n-1个向量()

B线性无关

14.设向量组a1.a2…as(s≥2)线性无关,且可由向量组B1.B2…Bs线性表示,则以下结论中

不能成立的是()

C存在一个aj,向量组aj,b2…bs线性无关

15.矩阵【10100…】的秩为()

A5

16.向量组a1.a2…as(s≥2)线性无关的充分必要条件是()

C a1.a2…as每一个向量均不可由其余向量线性表示

17.若线性方程组的增广矩阵为A=【1.~.2】则~=()时,线性方程组有无穷多解。

D1/2

18.a1.a2.a3是四元非齐次线性方程组AX=B的三个解向量,且

r(A)=3,a1=(1.2.3.4)T,a2+a3=(0.1.2.3)t,C表示任意常数,则线性方程组AX=B的通解X=() C(1.2.3.4)t+c(2.3.4.5)t

19.设a1.a2.a3是齐次线性方程组AX=0的基础解系,下列向量组不能构成AX=0基础解系

的是()

C a1-a2,a2-a3,a3-a1

20.AX=0是n元线性方程组,已知A的秩r<n,则下列为正确的结论是()

D该方程组有n-r个线性无关的解

21.方程组{x1-3x2+2x3=0…的一组基础解系是由()几个向量组成

B2

22.设m*n矩阵A的秩等于n,则必有()

D m≥n

23.一组秩为n的n元向量组,再加入一个n元向量后向量组的秩为()

C n

24.设线性方程组AX=B中,若r(A,b)=4,r(A)=3,则该线性方程组()

B无解

25.齐次线性方程组{X1+X3=0…的基础解系含()个线性无关的解向量。

B2

26.向量组a1.a2…as(s≥2)线性相关的充要条件是()

C a1.a2…as中至少有一个向量可由其余向量线性表示

27.设a1.a2是非齐次线性方程组AX=B的解,B是对应的齐次方程组AX=0的解,则AX=B

必有一个解是()

D B+1/2A1+1/2A2

28.齐次线性方程组{X1+X2+X3=0的基础解系所含解向量的个数为()

B2

1.设A为3*2矩阵,B为2*3矩阵,则下列运算中()可以进行

A AB

2.已知B1B2A1A2A3为四维列向量组,且行列式【A】=【a1,a2,a3,b1】=-4,【B】=【a1,a2,a3,B2】

=-1,则行列式【A+B】=()

D-40

3.设A为n阶非奇异矩阵(n>2),A为A的伴随矩阵,则()

A(A-1)+=【A】-1A

4.设A,B都是n阶矩阵,且AB=0,则下列一定成立的是()

A【A】=0或【B】=0

5.设A,B均为n阶可逆矩阵,则下列各式中不正确的是()

B(A+B)-1=A-1+B-1

6.设n阶矩阵A,B,C满足关系式ABC=E,其中E是n阶单位矩阵,则必有()

D BCA=E

7.设A是n阶方阵(n≥3),A是A的伴随矩阵,又k为常数,且k≠0,+-1,则必有(Ka)

+=()

B kn-1A+

8.设A是n阶可逆矩阵,A是A的伴随矩阵,则有()

A【A+】=【A】n-1

9.设A=【a11a12a13】,B=【a21a22a23】p1=【010】p2=【100】则必有()

C P1P2A=B

10.设A1B均为n阶方阵,则必有()

D【AB】=【BA】

11.设n维向量a=(1/2,0…0.1/2),矩阵A=E-ATA,B=E+2ATA,其中E为n阶单位矩阵,则AB=

()

C E

12.设A是n阶可逆矩阵(n≥2),A*是A的伴随矩阵,则()

C(A+)+=【A】n-2A

13.设A,B,A+B,A-1,+B-1均为n阶可逆矩阵,则(A-1+B-1)-1等于()

C A(A+B)-1B

14.设A,B为同阶可逆矩阵,则下列等式成立的是()

B(ABT)-1=(BT)-1A-1

15.设A为4阶矩阵且【A】=-2,则【【A】=()

C-25

16.设A=(1,2),B=(-1,3),E是单位矩阵,则ATB-E=()

D【-23】

17.下列命题正确的是()

D可逆阵的伴随阵仍可逆

18.设A和B都是n阶可逆阵,若C=(0B),则C-1=()

C(0A-1)

19.设矩阵A=【210】,矩阵B满足ABA+=2BA+E,其中E为三阶单位矩阵,A为A的伴随

矩阵,则【B】=()

B1/9

1.当k=()时,向量(

2.1.0.3)与(1.-1.1.k)的内积为2

C1/3

2.下列矩阵中,()是正交矩阵

C【3/5-4/5】

3.设a=(0,y,-1/2)t,B=(x,0,0)t它们规范正交,即单位正交,则()

B X≠+-1Y=+-1/2

4.若A是实正交方阵,则下述各式中()是不正确的

C【A】=1

5.下列向量中,()不是单位向量

C(0.1/2.-1/2)T

6.R3中的向量a=(2.3.3)t在基!1=(1.0.1)t,!2=(1.1.0)t!3=(0.1.1)t下的坐标为

B(1.1.2)

7.假设A,B都是n阶实正交方阵,则()不是正交矩阵。

D A+B

8.设a1=【200】,a2=【001】a3=【011】与!【100】!2【010】!3【001】是

R3的两组基,则()

B由基!1!2!3到基a1a2a3的过渡矩阵为【200】

1.若(),则A相似于B

D n阶矩阵A与B有相同的特征值,且n个特征值各不相同

2.n阶方阵与对角矩阵相似的充要条件是()

C矩阵A有n个线性无关的特征向量

3.A与B是两个相似的n阶矩阵,则()

A存在非奇异矩阵P,使P-1AP=B

4.设A=【124。。。】且A的特征值为1,2,3,则X=()

B4

5.矩阵A的不同特征值对应的特征向量必()

B线性无关

6.已知A=【31…】下列向量是A的特征向量的是()

B【-11】

7.三阶矩阵A的特征值1,0,-1,则f(A)=A2-2A-E的特征值为()

A-2.-1.2

8.设A和B都是n阶矩阵且相似,则()

C AB有相同的特征值

9.当n阶矩阵A满足()时,它必相似于对矩阵

C A有n个不同的特征值

10.设A是n阶实对称矩阵,则()

D存在正交矩阵P,使得PTAP为对角阵

11.设矩阵B=P-1AP,A的特征值~0的特征向量是a,则矩阵B的关于特征值~0的特征向量是

()

C P-1A

12.设A是n阶矩阵,适合A2=A,则A的特征值为()

A0或1

13.与矩阵A=【13.。。】相似的矩阵是()

B【10.。。】

14.A是n阶矩阵,C是正交矩阵,且B=CTAC,则下列结论不成立的是()

D A和B有相同的特征向量

15.n阶级方阵A与对角矩阵相似的充要条件是()

C矩阵A有n个线性无关的特征向量

16.已知A2=E,则A的特征值是()

C~=-1或~=1

17.设实对称矩阵A=【31。。。】的特征值是()

A【400…】

18.矩阵A=【31…】的特征值是()

C~1=-2~2=4

19.设~=2是非奇矩阵A的一个特征值,则矩阵(1/3A2)-1有一个特征值等于()

B3/4

20.n阶矩阵A具有n个不同的特征值是A与对角矩阵相似的()

C充分而非必要条件

21.矩阵A=【100…】与矩阵()相似

C A=【100…】

22.设A是n阶对称矩阵,B是n阶反对称矩阵,则下列矩阵中,不能通过正交变换化成对

角阵的是()

D ABA

1.二次型f(X1.X

2.X3)=X12-X22-2X32-6X1X3+2X2X3的矩阵为()

A【10-3…】

2.设矩阵A=(au)3*3,则二次型f(X1.X2.X3)=$(ai1x1+ai2x2+ai3x3)2的矩阵为()

C ATA

3.二次型XTAX经满秩线性变换X=CY化为变量为Y1.Y2…YN的二次型YTAX,则矩阵A

和B()

A一定合同

4.n阶实对称矩阵A合同于矩阵B的充分必要条件是()

D r(a)=r(b)且A与B的正惯性指数相等

5.设A为n阶非零矩阵,则()一定是某个二次型的矩阵

C ATA

6.矩阵A=【02/21…】对应的实二次型为()

C2X1X2+3X22+2X1X3-3X2X3

7.二次型f(x1.x2)=x12+6x1x2+3x22的矩阵表示为()

B(X1X2)【13…】【x1x2】

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答案B 解析 3. (1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案B 解析 4. (1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案A 解析 5.

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(1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案C 解析 8. (1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案B 解析 9.

(1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案B 解析 10. (1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案D 解析 11.

? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案C 解析 12. (1分) ? A. A ? B. B ? C. C ? D. D 纠错 得分:0 知识点:1 展开解析 答案C 解析 13.

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