全国2010年4月高等教育线性代数(经管类)自考试题
2011年1月-2012年4月自考04184线性代数(经管类)历年真题试题及答案

全国2012年4月高等教育自学考试线性代数(经管类)试题 课程代码:04184说明:在本卷中,A T表示矩阵A 的转置矩阵,A *表示矩阵A 的伴随矩阵,E 是单位矩阵,|A |表示方阵A 的行列式,r (A)表示矩阵A 的秩. 一、 单项选择题(本大题共10小题,每小题2分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1.设行列式111213212223313233a a a a a a a a a =2,则111213212223313233232323a a a a a a a a a ------=( )A.-12B.-6C.6D.122.设矩阵A =120120003⎛⎫ ⎪⎪ ⎪⎝⎭,则A *中位于第1行第2列的元素是()A.-6B.-3C.3D.63.设A 为3阶矩阵,且|A |=3,则1()A --=( )A.-3B.13-C.13D.34.已知4⨯3矩阵A 的列向量组线性无关,则A T 的秩等于( ) A.1B.2C.3D.45.设A 为3阶矩阵,P =100210001⎛⎫ ⎪⎪ ⎪⎝⎭,则用P 左乘A ,相当于将A ( )A.第1行的2倍加到第2行B.第1列的2倍加到第2列C.第2行的2倍加到第1行D.第2列的2倍加到第1列 6.齐次线性方程组123234230+= 0x x x x x x ++=⎧⎨--⎩的基础解系所含解向量的个数为( )A.1B.2C.3D.47.设4阶矩阵A 的秩为3,12ηη,为非齐次线性方程组Ax =b 的两个不同的解,c 为任意常数,则该方程组的通解为( ) A.1212cηηη-+ B.1212c ηηη-+ C.1212cηηη++ D.1212c ηηη++8.设A 是n 阶方阵,且|5A +3E |=0,则A 必有一个特征值为( ) A.53-B.35-C.35D.539.若矩阵A 与对角矩阵D =100010001-⎛⎫ ⎪- ⎪ ⎪⎝⎭相似,则A 3=( )A.EB.DC.AD.-E10.二次型f 123(,,)x x x =22212332x x x +-是( ) A.正定的B.负定的C.半正定的D.不定的二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
全国2010年4月高等教育自学考试线性代数(经管类)试题

全国2010年4月高等教育自学考试线性代数(经管类)试题全国2010年4月高等教育自学考试线性代数(经管类)试题课程代码:04184一、单项选择题(本大题共20小题,每小题1分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1.已知2阶行列式=m , =n ,则=()A.m-nB.n-mC.m+nD.-(m+n)2.设A , B , C均为n阶方阵,AB=BA,AC=CA,则ABC=()A.ACBB.CABC.CBAD.BCA3.设A为3阶方阵,B为4阶方阵,且行列式|A|=1,|B|=-2,则行列式||B|A|之值为()A.-8B.-2C.2D.84.已知A= ,B= ,P= ,Q= ,则B=()A.PAB.APC.QAD.AQ5.已知A是一个3×4矩阵,下列命题中正确的是()A.若矩阵A中所有3阶子式都为0,则秩(A)=2B.若A中存在2阶子式不为0,则秩(A)=2C.若秩(A)=2,则A中所有3阶子式都为0D.若秩(A)=2,则A中所有2阶子式都不为06.下列命题中错误的是()A.只含有一个零向量的向量组线性相关B.由3个2维向量组成的向量组线性相关C.由一个非零向量组成的向量组线性相关D.两个成比例的向量组成的向量组线性相关7.已知向量组α1,α2,α3线性无关,α1,α2,α3,β线性相关,则()A.α1必能由α2,α3,β线性表出B.α2必能由α1,α3,β线性表出C.α3必能由α1,α2,β线性表出D.β必能由α1,α2,α3线性表出8.设A为m×n矩阵,m≠n,则齐次线性方程组Ax=0只有零解的充分必要条件是A 的秩()A.小于mB.等于mC.小于nD.等于n9.设A为可逆矩阵,则与A必有相同特征值的矩阵为()A.ATB.A2C.A-1D.A*10.二次型f(x1,x2,x3)= 的正惯性指数为()A.0B.1C.2D.3二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
2011年1月-2012年4月自考04184线性代数(经管类)试题及答案

全国2012年4月高等教育自学考试线性代数(经管类)试题 课程代码:04184说明:在本卷中,A T表示矩阵A 的转置矩阵,A *表示矩阵A 的伴随矩阵,E 是单位矩阵,|A |表示方阵A 的行列式,r (A)表示矩阵A 的秩.一、 单项选择题(本大题共10小题,每小题2分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1.设行列式111213212223313233a a a a a a a a a =2,则111213212223313233232323a a a a a a a a a ------=( ) A.-12B.-6C.6D.122.设矩阵A =120120003⎛⎫⎪⎪ ⎪⎝⎭,则A *中位于第1行第2列的元素是( )A.-6B.-3C.3D.63.设A 为3阶矩阵,且|A |=3,则1()A --=( ) A.-3B.13-C.13D.34.已知4⨯3矩阵A 的列向量组线性无关,则A T 的秩等于( ) A.1B.2C.3D.45.设A 为3阶矩阵,P =100210001⎛⎫ ⎪⎪ ⎪⎝⎭,则用P 左乘A ,相当于将A ( )A.第1行的2倍加到第2行B.第1列的2倍加到第2列C.第2行的2倍加到第1行D.第2列的2倍加到第1列 6.齐次线性方程组123234230+= 0x x x x x x ++=⎧⎨--⎩的基础解系所含解向量的个数为( )A.1B.2C.3D.47.设4阶矩阵A 的秩为3,12ηη,为非齐次线性方程组Ax =b 的两个不同的解,c 为任意常数,则该方程组的通解为( ) A.1212cηηη-+ B.1212c ηηη-+ C.1212cηηη++ D.1212c ηηη++8.设A 是n 阶方阵,且|5A +3E |=0,则A 必有一个特征值为( )A.53-B.35-C.35D.539.若矩阵A 与对角矩阵D =100010001-⎛⎫ ⎪- ⎪ ⎪⎝⎭相似,则A 3=( )A.EB.DC.AD.-E10.二次型f 123(,,)x x x =22212332x x x +-是( ) A.正定的 B.负定的 C.半正定的 D.不定的二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
线性代数自考(经管类)

3.对行列式中有一行或一列中只有一个或两个非零元的情况,用这一行或一列展开.
4.行列式中各行元素之和为一个常数的类型.
5.范德蒙行列式的计算公式
例6求4阶行列式的值.
测试点 行列式的计算
解
测试点 个维向量线性无关相应的行列式;
解
所以 且.
答案 且.
2. 关于线性相关的几个定理
1) 如果向量组线性无关,而线性相关,则可由线性表示,且表示法唯一.
矩阵的加、减、乘有意义的充分必要条件
例1设矩阵,, ,则下列矩阵运算中有意义的是( )
A. B.
C. D.
测试点: 矩阵相乘有意义的充分必要条件
答案: B
例2设矩阵, ,则 =_____________.
测试点: 矩阵运算的定义
解 .
例3设矩阵, ,则____________.
3.转置 对称阵和反对称阵
1)转置的性质
2)若,则称为对称(反对称)阵
例4矩阵为同阶方阵,则=( )
A. B.
C. D.
答案: B
例5设令,试求.
测试点 矩阵乘法的一个常用技巧
解 因为,所以
答案
例6为任意阶矩阵,下列矩阵中为反对称矩阵的是( )
1.向量组的线性相关性的定义和充分必要条件:
1)定义: 设是一组维向量.如果存在个不全为零的数,使得
,
则称向量组线性相关,否则,即如果,必有
,则称向量组线性无关.
2) 个维向量线性相关的充分必要条件是至少存在某个是其余向量的线性组合.即线性无关的充分必要条件是其中任意一个向量都不能表示为其余向量的线性组合.
(完整版)历年全国自考线性代数试题及答案

浙02198# 线性代数试卷 第1页(共25页)全国2010年7月高等教育自学考试试卷说明:在本卷中,A T 表示矩阵A 的转置矩阵;A *表示A 的伴随矩阵;R (A )表示矩阵A 的秩;|A |表示A 的行列式;E 表示单位矩阵。
1.设3阶方阵A=[α1,α2,α3],其中αi (i=1,2,3)为A 的列向量, 若|B |=|[α1+2α2,α2,α3]|=6,则|A |=( )A.-12 B.-6 C.6 D.122.计算行列式=----32320200051020203( )A.-180 B.-120C.120 D.1803.设A =⎥⎦⎤⎢⎣⎡4321,则|2A *|=( )A.-8 B.-4C.4 D.8 4.设α1,α2,α3,α4都是3维向量,则必有 A. α1,α2,α3,α4线性无关 B. α1,α2,α3,α4线性相关 C. α1可由α2,α3,α4线性表示D. α1不可由α2,α3,α4线性表示5.若A 为6阶方阵,齐次线性方程组Ax =0的基础解系中解向量的个数为2,则R (A )=( )A .2 B 3C .4 D .56.设A 、B 为同阶矩阵,且R (A )=R (B ),则( )A .A 与B 相似B .|A |=|B |C .A 与B 等价D .A 与B 合同7.设A 为3阶方阵,其特征值分别为2,l ,0则|A +2E |=( )A .0 B .2C .3D .248.若A 、B 相似,则下列说法错误..的是( )A .A 与B 等价 B .A 与 B 合同C .|A |=|B | D .A 与B 有相同特征 9.若向量α=(1,-2,1)与β= (2,3,t )正交,则t =( )A .-2 B .0C .2D .410.设3阶实对称矩阵A 的特征值分别为2,l ,0,则( )A .A 正定 B .A 半正定C .A 负定D .A 半负定二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
自考 线性代数 04184 07年到10年全套真题

全国2010年7月高等教育自学考试线性代数(经管类)试题课程代码:04184试卷说明:在本卷中,A T 表示矩阵A 的转置矩阵;A *表示A 的伴随矩阵;r (A )表示矩阵A 的秩;| A |表示A 的行列式;E 表示单位矩阵。
一、单项选择题(本大题共10小题,每小题2分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1.设3阶方阵A =(α1,α2,α3),其中αi (i =1,2,3)为A 的列向量,若| B |=|(α1+2α2,α2,α3)|=6,则| A |=( ) A.-12 B.-6 C.6D.122.计算行列式32 3 20 2 0 0 0 5 10 20 2 0 3 ----=( )A.-180B.-120C.120D.1803.若A 为3阶方阵且| A -1 |=2,则| 2A |=( ) A.21B.2C.4D.84.设α1,α2,α3,α4都是3维向量,则必有( ) A.α1,α2,α3,α4线性无关 B.α1,α2,α3,α4线性相关 C.α1可由α2,α3,α4线性表示D.α1不可由α2,α3,α4线性表示5.若A 为6阶方阵,齐次线性方程组Ax =0的基础解系中解向量的个数为2,则r (A )=( ) A.2 B.3 C.4D.56.设A 、B 为同阶方阵,且r (A )=r (B ),则( ) A.A 与B 相似 B.| A |=| B | C.A 与B 等价D.A 与B 合同7.设A 为3阶方阵,其特征值分别为2,1,0则| A +2E |=( ) A.0 B.2 C.3D.248.若A 、B 相似,则下列说法错误..的是( )A.A 与B 等价B.A 与B 合同C.| A |=| B |D.A 与B 有相同特征值9.若向量α=(1,-2,1)与β=(2,3,t )正交,则t =( ) A.-2 B.0 C.2D.410.设3阶实对称矩阵A 的特征值分别为2,1,0,则( ) A.A 正定 B.A 半正定 C.A 负定 D.A 半负定二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
2011年1月-2012年4月自考04184线性代数(经管类)历年试题及答案 - 副本
2011年1月-2012年4月自考04184线性代数(经管类)历年试题及答案全国2012年4月高等教育自学考试线性代数(经管类)试题 课程代码:04184说明:在本卷中,A T表示矩阵A 的转置矩阵,A *表示矩阵A 的伴随矩阵,E 是单位矩阵,|A |表示方阵A 的行列式,r (A)表示矩阵A 的秩.一、 单项选择题(本大题共10小题,每小题2分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1.设行列式111213212223313233a a a a a a a a a =2,则111213212223313233232323a a a a a a a a a ------=( )A.-12B.-6C.6D.122.设矩阵A =120120003⎛⎫ ⎪⎪ ⎪⎝⎭,则A *中位于第1行第2列的元素是()A.-6B.-3C.3D.63.设A 为3阶矩阵,且|A |=3,则1()A --=( )A.-3B.13-C.13D.34.已知4⨯3矩阵A 的列向量组线性无关,则A T 的秩等于( ) A.1B.2C.3D.45.设A 为3阶矩阵,P =100210001⎛⎫ ⎪⎪ ⎪⎝⎭,则用P 左乘A ,相当于将A ( )A.第1行的2倍加到第2行B.第1列的2倍加到第2列C.第2行的2倍加到第1行D.第2列的2倍加到第1列18.设A 为3阶矩阵,且|A |=6,若A 的一个特征值为2,则A *必有一个特征值为_________. 19.二次型f 123(,,)x x x =2221233x x x -+的正惯性指数为_________.20.二次型f 123(,,)x x x =22212323224x x x x x --+经正交变换可化为标准形______________.三、计算题(本大题共6小题,每小题9分,共54分)21.计算行列式D =3512453312012034---- 22.设A =130210002-⎛⎫ ⎪⎪ ⎪⎝⎭,矩阵X 满足关系式A+X=XA ,求X. 23.设234αβγγγ,,,,均为4维列向量,A =(234αγγγ,,,)和B =(234βγγγ,,,)为4阶方阵.若行列式|A |=4,|B |=1,求行列式|A+B |的值.24.已知向量组1α=(1,2,-1,1)T ,2α=(2,0,t ,0)T ,3α=(0,-4,5,-2)T ,4α=(3,-2,t+4,-1)T (其中t 为参数),求向量组的秩和一个极大无关组.25.求线性方程组12341234123423222547x x x x x x x x x x x x +++=⎧⎪++-=⎨⎪+++=⎩的通解..(要求用它的一个特解和导出组的基础解系表示)26.已知向量1=α(1,1,1)T ,求向量23αα,,使123ααα,,两两正交.四、证明题(本题6分)27.设A 为m ⨯n 实矩阵,A T A 为正定矩阵.证明:线性方程组A x =0只有零解.全国2012年1月自考 《线性代数(经管类)》试题课程代码:04184说明:本卷中,A -1表示方阵A 的逆矩阵,r (A )表示矩阵A 的秩,||α||表示向量α的长度,αT表示向量α的转置,E 表示单位矩阵,|A |表示方阵A 的行列式.一、单项选择题(本大题共10小题,每小题2分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
月月自考线性代数(经管类)历试题及答案
全国2012年4月高等教育自学考试 线性代数(经管类)试题 课程代码:04184说明:在本卷中,A T表示矩阵A 的转置矩阵,A *表示矩阵A 的伴随矩阵,E 是单位矩阵,|A |表示方阵A 的行列式,r (A)表示矩阵A 的秩.一、 单项选择题(本大题共10小题,每小题2分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1.设行列式111213212223313233a a a a a a a a a =2,则111213212223313233232323a a a a a a a a a ------=( )A.-12B.-6C.6D.122.设矩阵A =120120003⎛⎫ ⎪⎪ ⎪⎝⎭,则A *中位于第1行第2列的元素是()A.-6B.-3C.3D.63.设A 为3阶矩阵,且|A |=3,则1()A --=( )A.-3B.13-C.13D.34.已知4⨯3矩阵A 的列向量组线性无关,则A T 的秩等于( ) A.1B.2C.3D.45.设A 为3阶矩阵,P =100210001⎛⎫ ⎪⎪ ⎪⎝⎭,则用P 左乘A ,相当于将A ( )A.第1行的2倍加到第2行B.第1列的2倍加到第2列C.第2行的2倍加到第1行D.第2列的2倍加到第1列6.齐次线性方程组123234230+= 0x x x x x x ++=⎧⎨--⎩的基础解系所含解向量的个数为( )A.1B.2C.3D.47.设4阶矩阵A 的秩为3,12ηη,为非齐次线性方程组Ax =b 的两个不同的解,c 为任意常数,则该方程组的通解为( )A.1212cηηη-+ B.1212c ηηη-+ C.1212cηηη++ D.1212c ηηη++8.设A 是n 阶方阵,且|5A +3E |=0,则A 必有一个特征值为( ) A.53-B.35-C.35D.539.若矩阵A 与对角矩阵D =100010001-⎛⎫ ⎪- ⎪ ⎪⎝⎭相似,则A 3=( ) A.E B.DC.AD.-E10.二次型f 123(,,)x x x =22212332x x x +-是( )A.正定的B.负定的C.半正定的D.不定的二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
4月自学考试 04184线性代数(经管类)试卷及答案
20XX 年4月高等教育自学考试全国统一命题考试04184 线性代数(经管类)试卷一、单项选择题(本大题共5小题,每小题2分,共10分)在每小题列出的四个备选项中只有一个选项是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
1、设行列式D 1=2211b a b a ,D 2=2221113232a b a a b a --,则D 2= 【 】A.-D 1B.D 1C.2D 1D.3D 12、若A=⎪⎪⎭⎫ ⎝⎛1x 1021,B =⎪⎪⎭⎫⎝⎛y 24202,且2A =B ,则 【 】A.x=1,y=2B.x=2,y=1C.x=1,y=1D.x=2,y=23、已知A 是3阶可逆矩阵,则下列矩阵中与A 等价的是 【 】A.⎪⎪⎪⎭⎫ ⎝⎛000000001B.⎪⎪⎪⎭⎫ ⎝⎛000010001C.⎪⎪⎪⎭⎫ ⎝⎛100000001D.⎪⎪⎪⎭⎫⎝⎛1000100014、设2阶实对称矩阵A 的全部特征值味1,-1,-1,则齐次线性方程组(E +A )x =0的基础 解系所含解向量的个数为 【 】 A.0 B.1 C.2 D.35、矩阵⎪⎪⎭⎫⎝⎛--3113有一个特征值为 【 】 A.-3 B.-2 C.1 D.2二、填空题(本大题共10小题,每小题2分,共20分)请在每小题的空格中填上正确答案。
错填、不填均无分。
6、设A 为3阶矩阵,且A =3,则13-A = .7、设A =⎪⎪⎭⎫ ⎝⎛5312,则A *= .8、已知A =⎪⎪⎭⎫⎝⎛1201,B =⎪⎪⎭⎫⎝⎛-211111,若矩阵X 满足AX =B ,则X = . 9、若向量组=1α(1,2,1)T ,=2α(k-1,4,2)T 线性相关,则数k= .10、若齐次线性方程组⎪⎩⎪⎨⎧=-+=+-=++030202321321321x x x x x x ax x x 有非零解,则数a = .11、设向量=1α(1,-2,2)T ,=2α(2,0,-1)T ,则内积(21,αα)= . 12、向量空间V ={x=(x 1,x 2,0)T |x 1,x 2R ∈}的维数为 .13、与向量(1,0,1)T 和(1,1,0)T 均正交的一个单位向量为 . 14、矩阵⎪⎪⎭⎫⎝⎛3221的两个特征值之积为 . 15、若实二次型f(x1,x2,x3)=2123222212x x x a ax x +++正定,则数a 的取值范围是 .三、计算题(本大题共7小题,每小题9分,共63分)16、计算行列式D =5111141111311112的值.17、设2阶矩阵A 的行列式21=A ,求行列式*12)2(A A +-的值.18、设矩阵A =⎪⎪⎪⎭⎫ ⎝⎛---101111010,B =⎪⎪⎪⎭⎫ ⎝⎛--301521,矩阵X 满足X =AX +B ,求X .19、求向量组T T T T )10,1,3(,)6,3,1(,)1,5,2(,)1,2,1(4321-=--===αααα的秩和一个极大线性无关组,并将向量组中的其余向量由该极大线性无关组线性表出.20、利用克拉默法则解线性方程组⎪⎩⎪⎨⎧=++=++=++232212322123221333c x c cx x b x b bx x a x a ax x ,其中c b a ,,两两互不相同.21、已知矩阵⎪⎪⎪⎭⎫ ⎝⎛=1111311a a A 与⎪⎪⎪⎭⎫ ⎝⎛=b B 00010000相似,求数b a ,的值.22、用正交变换化二次型212121455),(x x x x x x f ++=为标准型,并写出所作的正交变换.四、证明题(本题7分)23、设A ,B 均为n 阶矩阵,且A =B +E ,B 2=B ,证明A 可逆.20XX 年4月高等教育自学考试全国统一命题考试线性代数(经管类)试题答案及评分参考(课程代码 04184)一、单项选择题(本大题共5小题,每小题2分类,共10分)1.C2.A3.D4.C5.B 二、填空题(本大题共10小题,每小题2分,共20分)6. 97.⎪⎪⎭⎫⎝⎛--23158.⎪⎪⎭⎫⎝⎛--031111 9. 3 10. -2 11. 0 12. 2 13.()()T T 1,1,1311,1,131---或14. -1 15.a >1 三、计算题(本大题共7小题,每小题9分,共63分)16.解 D=40200320115011315111141111121131------=- (5分) =74402032115=-- (9分) 17.解 由于21=A ,所以A 可逆,于是1*-=A A A (3分) 故11*12212)2(---+=+A A A A A (6分) =2923232112111=⎪⎭⎫⎝⎛==+----A A A A (9分)18.解 由B AX X +=,化为()B X A E =-, (4分)而⎪⎪⎪⎭⎫ ⎝⎛--=-201101011A E 可逆,且()⎪⎪⎪⎭⎫ ⎝⎛--=--110123120311A E (7分)故⎪⎪⎪⎭⎫ ⎝⎛--=⎪⎪⎪⎭⎫ ⎝⎛--⎪⎪⎪⎭⎫⎝⎛--=11021335021111012312031X (9分)19.解 由于()⎪⎪⎪⎭⎫ ⎝⎛--→⎪⎪⎪⎭⎫ ⎝⎛----→00007510171101751075103121,,,4321αααα (5分) 所以向量组的秩为2,21,αα是一个极大线性无关组,并且有214213717,511αααααα-=+-= (9分)注:极大线性无关组不唯一。
2010年10月高等教育自学考试全国统一命题考试04184线性代数(经管类)试卷 (1)
2010年10月高等教育自学考试全国统一命题考试试卷04184线性代数(经管类)试卷一、单项选择题(本大题共10小题,每小题2分,共20分)1、设A 为3阶矩阵,A =1,则2T A -= ( ) A .-8 B. -2 C. 2 D. 82、设矩阵A =11⎛⎫⎪ ⎪-⎝⎭,B =(1,1),则AB = ( ) A .0 B.(1,-1) C.11⎛⎫⎪ ⎪-⎝⎭D.1111⎛⎫ ⎪ ⎪--⎝⎭3、设A 为n 阶对称矩阵,B 为n 阶反对称矩阵,则下列矩阵中为反对称矩阵的是( ) A .AB BA - B. AB BA + C. AB D. BA4、设矩阵A 的伴随矩阵*A =1234⎛⎫⎪⎝⎭,则1A -= ( )A .431212-⎛⎫- ⎪-⎝⎭ B. 121342-⎛⎫- ⎪-⎝⎭ C. 121342⎛⎫- ⎪⎝⎭ D. 421312⎛⎫- ⎪⎝⎭5、下列矩阵中不是初等矩阵的是 ( )A .101010000⎛⎫ ⎪⎪ ⎪⎝⎭ B.001010100⎛⎫⎪ ⎪ ⎪⎝⎭ C. 100030001⎛⎫⎪ ⎪ ⎪⎝⎭ D. 100010201⎛⎫⎪ ⎪ ⎪⎝⎭6、设,A B 均为n 阶可逆矩阵,则必有 ( ) A .A B +可逆 B.AB 可逆 C.A B -可逆 D.AB BA +可逆7、设向量组1α=(1,2), 2α=(0,2) β=(4,2),则 ( ) A .12,,ααβ线性无关 B .β不能由1α,2α线性表示C .β可由1α,2α线性表示,但表示法不惟一D .β可由1α,2α线性表示,且表示法惟一8、设A 为3阶实对称矩阵,A 的全部特征值为0,1,1,则齐次线性方程组()0E A x -= 的基础解系所含解向量的个数为 ( )A .0 B. 1 C. 2 D. 39、设齐次线性方程组1231231232000x x x x x x x x x λ-+=⎧⎪--=⎨⎪++=⎩有非零解,则λ为 ( )A .-1 B. 0 C. 1 D. 210、设二次型()Tf x X Ax =正定,则下列结论中正确的是 ( ) A .对任意n 维列向量x ,Tx Ax 都大于零 B .f 的标准形的系数都大于或等于零 C .A 的特征值都大于零 D .A 的所有子式都大于零二、填空题(本大题共10小题,每小题2分,共20分) 11、行列式0112的值为_________. 12、已知1223A ⎛⎫= ⎪⎝⎭,则A 中第一行第二列元素的代数余子式为_________.13、设矩阵1324A -⎛⎫=⎪-⎝⎭,1101P ⎛⎫= ⎪⎝⎭,则3AP =_________. 14、设,A B 都是3阶矩阵,且A =2,2B E =-,则1A B -=_________.15、已知向量组1α=(1,2,3),2α=(3,-1,2),3α=(2,3,k )线性相关,则数k=_________.16、已知Ax b =为4元线性方程组,() 3.r A = 1α,2α,3α为该方程组的3个解,且11234α⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭,233579αα⎛⎫ ⎪ ⎪+= ⎪ ⎪⎝⎭,则该线性方程组的通解是_________. 17、已知P 是3阶正交矩阵,向量132α⎛⎫ ⎪= ⎪ ⎪⎝⎭,102β⎛⎫⎪= ⎪ ⎪⎝⎭,则内积(,)P P αβ=_________.18、设2是矩阵A 的一个特征值,则矩阵3A 必有一个特征值为_________. 19、与矩阵1203A ⎛⎫=⎪⎝⎭相似的对角矩阵为_________.20、设矩阵122Ak-⎛⎫= ⎪-⎝⎭,若二次型Tf x Ax=正定,则实数k的取值范围是_________.三、计算题(本大题共6小题,每小题9分,共54分)21、求行列式0120101221010210D=的值.22、设矩阵010100001A-⎛⎫⎪= ⎪⎪⎝⎭,120210000B--⎛⎫⎪=-⎪⎪⎝⎭,求满足矩阵方程2XA B E-=的矩阵X.23、若向量组11 1 1α⎛⎫⎪= ⎪⎪⎝⎭,2113α⎛⎫⎪=- ⎪⎪⎝⎭,326kα⎛⎫⎪= ⎪⎪-⎝⎭,422kα-⎛⎫⎪= ⎪⎪-⎝⎭的秩为2,求k的值.24、设矩阵223110121A⎛⎫⎪=-⎪⎪-⎝⎭,21b⎛⎫⎪= ⎪⎪⎝⎭.(1)求1A-;(2)求解线性方程组Ax b=,并将b用A的列向量组线性表示.25、已知3阶矩阵A 的特征值为1-,1,2,设22B A A E =+-,求 (1)矩阵A 的行列式及A 的秩.(2)矩阵B 的特征值及与B 相似的对角矩阵.26、求二次型123121323(,,)422f x x x x x x x x x =-++经可逆线性变换112321233322222x y y y x y y y x y=++⎧⎪=-+⎨⎪=⎩所得的标准形.四、证明题(本题6分)27、设n 阶矩阵A 满足2A E =,证明A 的特征值只能是1±.。
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全国2010年4月高等教育线性代数(经管类)自考试题一、单项选择题(本大题共20小题,每小题1分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
一、单项选择题(本大题共20小题,每小题1分,共20分)在每小题列出的四个备选项中只有一个是符合题目要求的,请将其代码填写在题后的括号内。
错选、多选或未选均无分。
全国2010年4月高等教育综合英语(二)自考试题I. GRAMMARAND VOCABULARYComplete each of the following 15 sentences with the most likely answer. Write the letter corresponding to your choice on the answer sheet. (1 point each, 15 points in all)1. I was shocked by the number of people who seemed to have nothing better to do with their time than ______ to know fish better.A. getB. getsC. gettingD. got2. Not all birds can fly , ______ do all birds build nests.A. notB. noC. neverD. nor3. When you fall in love for the first time, it’s surprising ______ inconveniences you can put up with.A. thatB. whatC. whichD. those4. I know my limitations, so I won’t do anything ______ my ability.A. beyondB. underC. aboveD. below5. The hosts apologized for ______ that some of their guests might be vegetarians.A. having not consideredB. not having consideredC. having not been consideringD. not considering6. Mrs. Smith locked both the front and the back doors before she went to bed ______ that someone might steal into the house at night.A. soB. for fearC. unlessD. on condition7. We have to wait for the bus, ______?A. have weB. haven’t weC. do weD. don’t we8. ______ the popularity of robots in Japan, the country’s robot-production growth is slowing down.A. AsB. AlthoughC. WithD. Despite9. The proposal deserves support as it gives ______ to the needs of children.A. curiosityB. awarenessC. priorityD. emphasis10. Those familiar with evening dress will always ______ black with elegance.A. linkB. associateC. connectD. relate11. If you’re sitting around a lonely campfire at night, the howl of a wolf can sound pretty ______.A. fearfulB. threateningC. dreadfulD. frightening12. The suspect cooperated fully with the police when ______ about his role in the incident.A. requiredB. questionedC. demandedD. negotiated13. Homing pigeon is useful as a ______ carrier because when taken from home, it will return at the first opportunity.A. meaningB. informationC. messageD. content14. One of the responsibilities of the Coast Guard is to make sure that all ships ______ follow traffic rules in busy harbors.A. convenientlyB. obedientlyC. sufficientlyD. reluctantly15. She was so ______ the noise that she sent the children to bed.A. broken down byB. filled in withC. taken over byD. fed up withII. CLOZEFill in each of the 15 blanks in the passage with the most likely answer. Write the letter corresponding to your choice on the answer sheet. (1 point each, 15 points in all)A quarter of the world’s population now speaks English. If you want to 16 your products, or yourself, you have no choice 17 to do it in English. Politicians and business people must speak English if they want to make their mark 18 the world’s stage. Writers of minority languages can 19 hope to sell their books unless they write in English.As English is spoken all over the world, it means that no individual country can really exercise a 20 influence over it. In the past, it was the mother tongue 21 who controlled the future of the language. Now, indeed for the last few decades, the mother tongue users are in a significant 22 .This means that the character of the language could well 23 with new words, new rhythms, and new pronunciations. It isn’t going to be British and American English anymore- that’s 24 . All over the world education authorities are struggling to find the resources to meet the 25 for English.However, it is open to question whether they are putting all their eggs in one 26 . The future of English language is intimately 27 the electronic revolution. Satellite television and the Internet have helped accelerate the 28 of English, but will that always be the case?The Internet, 29 , is now seen as the saviour of minority languages. Indeed, with the continuing improvements in electronic translation, will we even need a(n) 30 language? It is open to question.16. A. sell B. make C. produce D. trade17. A. and B. but C. rather D. or18. A. on B. with C. for D. at19. A. ever B. still C. no longer D. no more20. A. negative B. typical C. subtle D. dominant21. A. speakers B. followers C. believers D. interpreters22. A. group B. number C. amount D. minority23. A. alter B. differ C. shift D. vary24. A. old B. new C. history D. future25. A. want B. pursuit C. demand D. desire26. A. box B. basket C. bag D. container27. A. restricted to B. drawn to C. tied up with D. caught up with28. A. range B. scope C. spread D. coverage29. A. as a result B. as a rule C. in addition D. in fact30. A. global B. unique C. worldly D. internationalⅢ. PARAPHRASINGChoose the closest paraphrased version after each of the following sentences or the italicized part. Write the letter corresponding to your choice on the answer sheet. (1 point each, 10 points in all)31. By definition, heroes and heroines are men and women distinguished by uncommon courage, achievements, and self-sacrifice made most often for the benefit of others - they are people against whom we measure others.A. ... we disagree with them when judging others.B. ... we think more highly of them than others do.C. ... we believe other people are different from them.D. ... we use them as the standard when evaluating others.32. Many people have a rather acute sense of the shortness of each lifetime.A. Many people realize that they are short of time.B. Many people feel keenly that their life is short.C. Many people are sensible only for a short period of time.D. Many people would like very much to prolong their life.33. As a scientist, she learned with sadness that little in Nature is truly beyond the tampering reach of man.A. … it is difficult for man to compete with Nature.B. … it is impossible for man to benefit from Nature.C. … Nature can hardly escape from man’s interference.D. …Nature is really more powerful than human beings.34. These bits of information and the list of D grades were all the records had to offer.A. These bits of information explained his poor grades.B. His poor grades were related to these bits of information.C. What could be found in the records were those things required to be there.D. The records gave nothing more than these bits of information and poor grades.35. People were only shapes in dense, gray fog of dust and ash.A. The dusty air made people barely visible.B. The air was too polluted for people to breathe.C. People hardly stayed in shape in such dirty air.D. People were almost buried in thick dust and ash.36. It would not be difficult to compile a list of such surprises that would fill the next fifty pages, but I will content myself with suggesting the first few that occur to me.A. ... I feel satisfied with the first few examples in my mind.B. ... I am surprised with the number of suggestions I can give.C. ... the first few examples are the best ones that satisfy my curiosity.D. ... the first few surprises coming to me would be enough as examples.37. If I love you, I can see you as a separate person, with your own values and thoughts and feelings, and I do not insist that you surrender your identity and conform to an image of what I expect you to be for me.A. ... I don’t want you to lose hope and then blame me for it.B. ... I won’t force you to be independent and you can rely on me.C. ... I don’t expect you to follow my step and become similar to me.D ... I won’t push you to lose yourself and become what I like you to be.38. If disappointed, though, she [my mother] wasted no energy on self-pity. She would make me make something of myself whether I wanted to or not.A. Dissatisfied as she was, she remained as determined as before.B. Faced with discouragements, she never lost her energetic spirits.C. Nothing could let her down even though no one sympathized with her.D. There was enough self-confidence in her to deal with disappointments.39. My maternal grandfather, it is true, was cut off in the flower of his youth at the age of sixty-seven.A. My maternal grandfather retired from gardening at sixty-seven.B. My maternal grandfather died young, when he was sixty-seven.C. My maternal grandfather lost energy when he was sixty-seven.D. My maternal grandfather was considered no longer young at sixty-seven.40. Far-reaching as many of them [Edison’s inventions] have been in their effect upon modern civilization, the total effect of Edison’s career surpasses the sum of all of them.A. One can never evaluate Edison’s great inventions high enough.B. One needs to understand Edison’s inventions to appreciate his career.C. Edison’s inventions are his most outstanding contributions to human life.D. Edison’s influence upon human society is much greater than his inventions.IV. READING COMPREHENSIONRead the two passages and choose the most likely answer to each of the questions. Write the letter corresponding to your choice on the answer sheet (2 points each, 20 points in all)Passage 1Friendship and friends continue to remain central to our lives. The relationship that we share with our friends is grounded in a mutual concern as our friends help us in shaping up our personalities as well. Even in this age of online social networking, the need of expanding the network of friends, whether online or offline remains a primary concern for people. We may have a very vague understanding of what makes a friend but we all want to have a good many friends around us.Research has shown that the quality and nature of your friends are one of the key influencing factors in achieving happiness and satisfaction. Friends often affect the health and energy of a person as well. More and more people are increasingly turning towards their friends for support and mutual sharing rather than communicating to their relatives; this sociological phenomenon has escalated the desire to make more friends. Though friendship is a relationship developed between two individuals marked by a voluntary, emotional and mutual concern, there are certain steps, which, if followed, can win you friends or at least help you get close to those whom you want to make friends with.The first step to make friends with someone is to make them like you. If they don’t like you in the first place, it is unlikely that they’ll be eager to become your friends. To let someone know that you are genuinely interested in them, simple gestures like a little smile and calling him by his first name can lay the basic foundation. To make the other person feel important you need to be a good listener and encourage the other person to talk. Give your honest and sincere opinion but do not overtly criticize or make fun of him or her.The second step of making someone your friend is to develop a mutual consideration between each other. You need to share his or her perspective and for that it is necessary for you to develop a tendency to see things from the other person’s point of view. To become best of friends you must show a genuine concern and consideration for your friend’s desires and opinions.The third and final step towards friendship is to show your unconditional support and encouragement towards your friend. Moreover, you also need to be very clear about your expectations from your friend. If what you expect from the person you want to befriend matches with what that person can and want to contribute in friendship, the developing friendship between you and your friend is assured to grow, making both of you gratified with mutual emotional bonding.Friendship has overwhelming significance on both social and individual levels. Friendship doesn’t only provide us with emotional as well as practical support but it also helps us improve our personal identities. So it is quite natural to feel the urge to reach out and make new friends, expanding your friend’s circle. Friendship remains familiar yet quite ambiguous (模糊的) for our deeper understanding, but then again, friends remain the most important ingredients in the recipe of life!41. The relationship between friends is based on ______.A. the social networking around friendsB. friends’ caring and helping each otherC. the style friends choose to communicateD. an unclear understanding of what makes a friend42. Which of the following statements has been proved correct?A. The better quality your friends have, the happier you will be.B. Communication with friends is healthier than that with relatives.C. The more friends you have, the more satisfied you are with yourself.D. Those you like in the first place are more likely to become your friends.43. The word “escalated” in the second paragraph probably means ______.A. intensifiedB. challengedC. shiftedD. resisted44. It can be inferred from the first suggested step that ______ is NOT likely to win you friends.A. making others talkB. being very talkativeC. telling people you like themD. addressing others in an informal way45. What structure does the author use to organize this passage?A. Effect- cause.B. Comparison- contrast.C. General- specific- general.D. Definition - explanation - example.Passage 2Every single student that applies for a class or course online has certain expectations and they are often extremely high. Unfortunately for most individuals, they will be disappointed. It is a fact that the majority of individuals believe that they will actually enjoy every aspect of their course. The truth is that there is very little in a degree to inspire until an individual actually completes the course and is awarded the degree or certificate that could change his or her life. This is no different with an online education and, in fact, the disappointment may be more obvious because they may face extra difficulties in an online course.Online education expectations are often incredibly high, as previously stated, in terms of what an individual can get from it as well as the manageability of the content itself. However, although an individual will ultimately be proven right about what he or she can get from it, this will not be the case for manageability. Most online education courses are anything but manageable for those who decide to take them. The level of work would be the same as any other degree or qualification course, but an individual is expected to manage it on his or her own instead of having a tutor or lecturer there to remind him or her about assignments and deadlines. Time management is extremely important, but the nature of these individuals’ life often makes it impossible to manage without feeling the stresses and strains of the burden.It is important to be realistic with your expectations concerning an online education course, but many are completely unrealistic. If you read any of the accounts from students that have already taken and may have even passed an online education course, many will speak in glowing terms; but you should also read between the lines.They claim that their expectations will ultimately be fulfilled in terms of the long-term benefits. However, in the short term they may say they found it difficult to cope with the workload, the lack of time that they actually had to complete it, an unsupportive employer and so on. This last point may not be an issue if you are doing it to advance yourself within your current workplace, but there will otherwise be some sort of resentment bubbling under the surface and that can indeed make your life a misery for a little while. If you expect it and prepare for it then it should not bother you too much, and if the resentment does not come then so much the better!In terms of your own expectations, the best attitude to have is not to go in with any at all. If you do not expect anything then you may be pleasantly surprised. It can be difficult to fit an online education in with personal commitments, such as a family to look after or a job to go to every day whilst trying to study. It does take a lot of effort, but if you do not expect itto be easy or expect that you will just be able to do it without the balance of your life being upset as a result, then you will probably find that you do get a sense of satisfaction from it! Similarly, if you expect the worst then you will probably be pleasantly surprised! Managing your expectations can be difficult, but do not let them get the better of you !46. The manageability is more difficult in online courses than in any other course because ______.A. online courses require higher level of workB. online courses bring much more assignmentsC. online course takers have no way to contact their lecturersD. online course takers have to rely on themselves for management47. According to the passage, one may not have the problem of an unsupportive employer if ______.A. he is fully prepared to deal with any resentment from his bossB. his online education will be beneficial to his current workplaceC. his expectations will be fulfilled in the form of long-term benefitsD. he can discuss the issue width his boss openly and bravely on the table48. According to the writer of this passage, one’s satisfaction in online education might come from ______.A. high expectationsB. diligent effortsC. a realistic attitudeD. a proper balance49. In the last sentence of this passage, the phrase “get the better of you” probably means “______”.A. make a fool of youB. take advantage of youC. get too fancy for you to imagineD. become too high for you to control50. This passage is written mainly for the purpose of ______.A. inspiring its readers to fulfill their expectations through hard workB. persuading its readers to be more practical with their personal commitmentsC. informing its readers about the disadvantages of the courses they take onlineD. preparing its readers for the hardships they may encounter in online educationV. WORD DERIVATIONComplete each of the following sentences with a (compound) word derived from the one(s) given in brackets. Write your word on the answer sheet. (1 point each, 10 points in all)51. The railway station has always been a fascinating place, representing departure as well as ______. (arrive)52. A climate that is extreme in temperature, whether it be too hot or too cold, is not ______. (desire)53. Tropical rainforest is a forest of tall trees in a region of year-round ______ and plentiful rainfall. (warm)54. We often hear of people breaking down from ______, but in nine cases out of ten they are really suffering from worry or anxiety. (work, over)55. The ______ red strawberries make a nice decoration on top of the cake. (juice)56. More and more businessmen have realized that it is important to ______ their products in international trade. (standard)57. The earthquake-stricken area received ______ help from its neighboring provinces. (time)58. What is considered moral in one society may be ______ in another, depending on different social values. (moral)59. Young people will need to learn to ______ the responsibility of their own decisions. (shoulder)60. Special guidance and help need to be provided to the ______ members of the staff. (qualified)VI. SENTENCE TRANSLATIONTurn the following sentences into English and write your sentences on the answer sheet. (3 points each, 15 points in all)61.公司越来越多地借助高科技来保护计算机的安全。