第一篇：2012年4月自考線性代數(shù)真題及答案

全國2012年4月高等教育自學考試線性代數(shù)(經(jīng)管類)試題課程代碼：04184

一、單項選擇題（本大題共10小題，每小題2分，共20分）

a111.設行列式a21a12a22a32a13?a112a122a222a32?3a13?3a23=（）?3a33D.12 a31A.-12 a23=2,則?a21a33?a31B.-6

C.6 ?120???2.設矩陣A=?120?,則A*中位于第1行第2列的元素是（）?003???A.-6 B.-3

C.3

D.6

3.設A為3階矩陣，且|A|=3，則(?A)?1=（）A.?3 B.?1 3C.1 3D.3 4.已知4?3矩陣A的列向量組線性無關，則AT的秩等于（）A.1 B.2

C.3

D.4 ?100???5.設A為3階矩陣,P =?210?,則用P左乘A，相當于將A（）?001???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.齊次線性方程組?A.1

?0?x1?2x2?3x3的基礎解系所含解向量的個數(shù)為（）?x2+x3?x4= 0?B.2

C.3

D.4 7.設4階矩陣A的秩為3，?1，?2為非齊次線性方程組Ax =b的兩個不同的解，c為任意常數(shù)，則該方程組的通解為（）A.?1?c?1??22 B.?1??223 5?c?1 C.?1?c?1??22 D.?1??225 3?c?1

8.設A是n階方陣，且|5A+3E|=0，則A必有一個特征值為（）A.?5 3B.?C.5D.??100???9.若矩陣A與對角矩陣D=?0?10?相似，則A3=（）?00?1???A.E B.D

C.A

D.-E

22210.二次型f(x1,x2,x3)=3x1是（）?2x2?x3A.正定的 B.負定的 C.半正定的 D.不定的

二、填空題（本大題共10小題，每小題2分，共20分）請在每小題的空格中填上正確答案。錯填、不填均無分。

111.行列式21146=____________.41636?001??100?????12.設3階矩陣A的秩為2，矩陣P =?010?，Q =?010?，若矩陣B=QAP，則r(B)=_____________.?100??101?????13.設矩陣A=??1?4??48?，B=???，則AB=_______________.??14??12?14.向量組?1=(1,1,1,1),?2=(1,2,3,4),?3=(0,1,2,3)的秩為______________.15.設?1,?2是5元齊次線性方程組Ax =0的基礎解系，則r(A)=______________.?10002???16.非齊次線性方程組Ax =b的增廣矩陣經(jīng)初等行變換化為?01002?，則方程組的通解是______.?0012-2???17.設A為3階矩陣，若A的三個特征值分別為1，2，3，則|A|=___________.18.設A為3階矩陣，且|A|=6，若A的一個特征值為2，則A*必有一個特征值為_________.22219.二次型f(x1,x2,x3)=x1的正慣性指數(shù)為_________.?x2?3x322220.二次型f(x1,x2,x3)=x1?2x2?2x3?4x2x3經(jīng)正交變換可化為標準形______________.三、計算題（本大題共6小題，每小題9分，共54分）

35?12?453?321.計算行列式D =

120120?34?1?30???22.設A=?210?，矩陣X滿足關系式A+X=XA,求X.?002???23.設?，?，?2，?3，?4均為4維列向量，A=（?，?2，?3，?4）和B=（?，?2，?3，?4）為4階方陣.若行列式|A|=4，|B|=1，求行列式|A+B|的值.24.已知向量組?1=(1,2,?1,1)T,?2=(2,0,t,0)T,?3=(0,?4,5,?2)T,?4=(3,?2,t+4,-1)T（其中t為參數(shù)），求向量組的秩和一個極大無關組.?x1?x2?2x3?x4?3?25.求線性方程組?x1?2x2?x3?x4?2的通解..（要求用它的一個特解和導出組的基礎解系表示）

?2x?x?5x?4x?734?1226.已知向量?1=(1,1,1)T，求向量?2，?3,使?1，?2，?3兩兩正交.四、證明題（本題6分）

27.設A為m?n實矩陣，ATA為正定矩陣.證明：線性方程組Ax=0只有零解.全國2012年4月高等教育自學考試線性代數(shù)(經(jīng)管類)試題答案課程代碼：04184

一、單項選擇題（本大題共10小題，每小題2分，共20分）

1．D 2.A 3.B 4.C 5.B 6.B 7.A 8.B 9.D 10.D

二、填空題（本大題共10小題，每小題2分，共20分）請在每小題的空格中填上正確答案。錯填、不填均無分。

? 2?? 0?? 2?? 0??00?11.16 12.213.?14.2 15.3 16.???k??,k為任意常數(shù) 17.6 ???2???2??00?

???? 0??? 1?2218.3 19.220.y1?4y2

三、計算題（本大題共6小題，每小題9分，共54分）

3?421.解：D =125?12153?3?4??20130?342201120153?301331???5?120?1?1?10?340?4?3212010?1?1?1

013310?4?3212010?1?1?1?10?12????48 00?10?1216001622.解：由A?X?XA，可知X(A?E)?A，則X?A(A?E)?1，?0?0?30?????1且A?E??200?，(A?E)???13?001??0???0??00? 01??120?0??11?1?30??0122??1?????100???10故X?A(A?E)??210???13??3?

?002??001??002???????23.解: A?B?(???,2?2,2?3,2?4)?8[(?,?2,?3,?4)?(?,?2,?3,?4)]?8A?B?40

???1?2?24.解：（?1,?2,?3,?4）=??1??1?1?0???0??0203??1203??12????0?4?2??0?4?4?8??01??t5t?4??0t?25t?7??0t?2????0?2?1??0?2?2?4??00203??1??112??0?03?t3?t??0??000??00?2?1??112?

03?t3?t??000?03??12?

5t?7??00?t?3時，秩為2，一個極大無關組為?1,?2 t?3時，秩為3，一個極大無關組為?1,?2,?3.25.解：對增廣矩陣作初等行變換

?11213??11213??11213???????A?(A,b)??121?12???01?1?2?1???01?1?2?1?

?21547??0?1121??00000????????10334???

??01?1?2?1?

?00000????x1??3x3?3x4?4同解方程組為?.x3，x4是自由未知量，特解?*?(4,?1,0,0)T

?x2?x3?2x4?1?x1??3x3?3x4導出組同解方程組為?.x3，x4是自由未知量，x?x?2x34?2基礎解系?1?(?3,1,1,0)T，?2?(?3,2,0,1)T，通解為???*?k1?1?k2?2,k1,k2?R.26.解：設?2=(x1,x2,x3)T，?2與?1正交，則有x1?x2?x3?0，故可取?2==(1,0,-1)T, 設?3=(y1,y2,y3)T，?3與?1，?2兩兩正交，則?故可取?3=(1,?2,1).四、證明題（本題6分）

27.證明：由于ATA為正定矩陣，則秩（ATA）= n，又秩（A）= 秩（ATA）= n，則線性方程組Ax=0只有零解.T?y1?y2?y3?0.?y1?y3 = 0

第二篇：0907線性代數(shù)真題及答案

全國2009年7月高等教育自學考試

線性代數(shù)（經(jīng)管類）試題

課程代碼：04184 試卷說明：在本卷中，AT表示矩陣A的轉置矩陣；A*表示A的伴隨矩陣;R(A)表示矩陣A的秩；|A|表示A的行列式；E表示單位矩陣。

一、單項選擇題(本大題共10小題，每小題2分，共20分)在每小題列出的四個備選項中只有一個是符合題目要求的，請將其代碼填寫在題后的 括號內。錯選、多選或未選均無分。

1.設A,B,C為同階方陣，下面矩陣的運算中不成立的是(C)．．．A.（A+B）T=AT+BT C.A(B+C)=BA+CA a112.已知a21a31a12a22a32a132a11a23=3，那么a21a33?2a312a12a22?2a32B.|AB|=|A||B| D.(AB)T=BTAT 2a13a23=(B)?2a33A.-24 C.-6

B.-12 D.12

3.若矩陣A可逆，則下列等式成立的是（C)A.A=1A* AB.A?0

C.(A2)?1?(A?1)2 D.(3A)?1?3A?1

?41?31?2???02?1???23?4.若A=?，B=，C=2矩陣的??3?12?，則下列矩陣運算的結果為3×??152??????21??是(D)A.ABC C.CBA

B.ACTBT D.CTBTAT

5.設有向量組A：?1，?2，?3，?4，其中?1，?2，?3線性無關，則(A)A.?1，?3線性無關

C.?1，?2，?3，?4線性相關

B.?1，?2，?3，?4線性無關 D.?2，?3，?4線性相關

浙04184# 線性代數(shù)（經(jīng)管類）試題

6.若四階方陣的秩為3，則（B)A.A為可逆陣

C.齊次方程組Ax=0只有零解

B.齊次方程組Ax=0有非零解 D.非齊次方程組Ax=b必有解

7.設A為m×n矩陣，則n元齊次線性方程Ax=0存在非零解的充要條件是(B)A.A的行向量組線性相關 C.A的行向量組線性無關

8.下列矩陣是正交矩陣的是（A)0??10?0?10A.???

??00?1???101?1??110B.?2???011??B.A的列向量組線性相關 D.A的列向量組線性無關

?cos?C.???sin??sin??

cos???????D.?????2202216661063??3?3??

3??3??3??9.二次型f?xTAx(A為實對稱陣)正定的充要條件是（D)A.A可逆

C.A的特征值之和大于0

0??k0??10.設矩陣A=?0k?2?正定，則(C)??0?24??B.|A|>0

D.A的特征值全部大于0

A.k>0 C.k>1 ??1?:D1?k?0;?2?:D2?k00k0?k2?0;0kB.k?0 D.k?1 ?3?D3??k?1k?20k?2?k?k?4k?4??4k?k?1??0

?240?2

4二、填空題(本大題共10小題，每小題2分，共20分)請在每小題的空格中填上正確答案。錯填、不填均無分。

浙04184# 線性代數(shù)（經(jīng)管類）試題

1??21??1??2????????6?3.11.設A=（1，3，-1），B=（2，1），則ATB=?63?。

ATB??3??2,?1??1???2????2?1?1??????21012.若131?0,則k?-1。

k2121021021131?131??????2?k?1??k?1?0.?k??1.k?1?1k21k?1?10?120??0?60???*??630?。20013.設A=?，則A=????2?1?4?????013??120121*?A?200?3??12?0.?A可逆.?A?1?A20A013?1??A*?AA?1.而?A,E3???2?0?0?120?11??②?4?????010?12-14?013?00?1?③320?100??1?②+(-2)?①?00?010???????0?013?001???0?①+(-2)?②?100?0?③+(-1)?②?0???????010?12?003??121???20?100???40??210?13?001??120??-140?141??

120?120??100?0?0?0?60?1???????????010?12-140?.?A?1??12-140?????630?12??001??1611213???1611213???????2?1?4??0?60??0?60?1???????A*?AA?1??12??????630????630?.?12??????2?1?4??2?1?4?14.已知A2-2A-8E=0，則（A+E）-1=?A2?2A?8E?0.??1?A?3E?。5?A2?2A?3E??5E?0?1?5??A?E??A?3E??5E.?11?A?E??A?3E??E?5?A?3E?A?E???????E.??A?E?? 1?A?3E?5浙04184# 線性代數(shù)（經(jīng)管類）試題

15.向量組?1?(1,1,0,2),?2?(1,0,1,0),?3?(0,1,?1,2)的秩為___2。

將三個向量的轉置向量拼成一個4?3的矩陣，化簡此矩陣?1?1A???0??210??110??11???③+1?②?+(-1)?①01?②0?110?1④+(-2)?①④+(-2)?②??????????????01?1??001?1?????02?0?22???000??1?.?r?A??2.?0?0?16.設齊次線性方程Ax=0有解?，而非齊次線性方程且Ax=b有解?,則???是方程組____Ax=b的解。

?x1?x2?0??17.方程組?的基礎解系為?1???1?。

?x2?x3?0?1???___1。18.向量??(3,2,t,1),??(t,?1,2,1)正交,則t?__________?t????1???,????3,2,t,1????3t?2?2t?1?t?1?0.?t?1.?2????1??1?

?10??3b?119.若矩陣A=?與矩陣B=相似，則x=?4?ab?。???3?04??ax?A?B?103b1??4?3x?ab?x??4?ab?.04ax312?32??1??22220?。20.二次型f(x1,x2,x3)?x1?2x2?3x3?x1x2?3x1x3對應的對稱矩陣是?12??320?3???

三、計算題（本大題共6小題，每小題9分，共54分）1?340403521.求行列式D=的值。

202?276?22

浙04184# 線性代數(shù)（經(jīng)管類）試題

1?3404035解D?202?276?22④+2?①1?340209040352?2624351?2??1???3?22?2按

解：把所有行向量轉置為列向量形成4?4的矩陣，并將其化為簡化階梯形矩陣.?23?54???11?53?????0?26?40?26?4③?①TTT?????A???1T,?2,?3,?4???????11?53??23?54?????3?19?53?19?5??????11?53???11?53???1?②??③+2?①0?26?40?13?2④+3?①2?????????????05?1510??05?1510?????02?6402?64??????10?21??102?1?①+1?②???1?①??③+5?②0?13?201?32④+2?②?1?②?????記為?,?,?,??B.???????1342?0000??0000?????00000000????顯然B是A的簡化階梯形矩陣.易見：B的秩為2，??從而A的秩為2,原向量組的秩為2.易見：B的列向量組的一個極大線性無關組為??1,?2?.T?A的列向量組的一個極大線性無關組為??1T,?2?;從而??1,?2?是原向量組的一個極大線性無關組.24.求?取何值時,齊次方程組 ?(??4)x1?3x2?0?

?4x1?x3?0

??5x??x?x?0123?

有非零解？并在有非零解時求出方程組的通解。

浙04184# 線性代數(shù)（經(jīng)管類）試題

解?方程的個數(shù)與未知量的個數(shù)相同，?考察系數(shù)矩陣A是否為可逆矩陣.??43A?4?50?01?1??430③+1?②4?101?0按

63????1??解?A的特征方陣?En?A??0??53?.?A的特征方程為?0?6??4?????1?En?A?0063??532??53=??-1????-1?????-1???2???2????-1????2??0???5???-4??18????6??4?6??4得?1=?2=1，?3=-2，為A的兩個特征值.用來求特征向量的矩陣方程為63??x1??0?????1?x1?6x2?3x3?0???1??????0??53x?0,即齊次線性方程組??E3?A?x?????5?x2?3x3?0.?2????????0??x??0??6??4???6x2????4?x3?03?????屬于?1??2?1的特征向量滿足線性方程組6x2?3x3?0,即x3??2x23個未知量1個方程,必有2個自由未知量,不妨取x1、x2為自由未知量，?1??0??x1??1??0?????令?????或??，則x3?0或?2，于是得2個線性無關的特征向量p1??0?，p2??1?.?x2??0??1??0???2???????3x1?6x2?3x3?0?屬于?3??2的特征向量滿足線性方程組為?3x2?3x3?0.,即x1?x2??x3,??6x2?6x3?0?3個未知量2個方程,必有1個自由未知量,不妨取x1為自由未知量，令x1?1,則x2?1,x3??1,?1???于是得1個線性無關的特征向量p3??1?.??1???22226.用配方法求二次型f(x1,x2,x3)?x1?4x2?x3?2x1x3?4x2x3的標準形，并寫出相應的線性變換。

222解?二次型f(x1,x2,x3)?x12?4x2?x3?2x1x3?4x2x3??x12?2x1x3?x3???4x22?4x2x3?x32??x322??x1?x3???2x2?x3??x322?x1?y1?y3?y1?x1?x3?1??設?y2?2x2?x3，即?x2??y2?y3?，2?y?x?33?x3?y3??222可使得f(x1,x2,x3)??x1?x3???2x2?x3??x3?g(y1,y2,y3)?y12?y2?y3.即二次型的標準形；221??y1??x1??10??????此時相應的線性變換x?Py為?x2???012?12??y2?.?x??00?1???3????y3?

浙04184# 線性代數(shù)（經(jīng)管類）試題

四、證明題（本大題共1小題，6分）

27.證明：若向量組?1,?2,??n線性無關,而?1??1??n,?2??1??2,?3??2??3,?，?n??n?1+?n，則向量組?1,?2,?,?n線性無關的充要條件是n為奇數(shù)。

證?設k1?1?k2?2???kn?n?0.將已知條件代入得k1??1??n??k2??1??2??k3??2??3????kn??n?1??n??0.整理得?k1?k2??1??k2?k3??2????kn?1?kn??n?1??kn?k1??n?0.??1,?2,??n線性無關,??k1?k2???k2?k3???kn?1?kn???kn?k1??0.?k1??k2?k3??k4????kn?k1,當n為奇數(shù)，則n?1為偶數(shù),則上式為?k1??k2?k3??k4????kn?1?kn??kn?k1.由此?kn??kn?0,?k1?k2?k3?k4???kn?1?kn?0.因此,?1,?2,?,?n線性無關.?反之,若?1,?2,?,?n線性無關,即當且僅當k1?k2?k3?k4???kn?1?kn?0時,等式k1?1?k2?2???kn?n?0才成立,?k1??1??n??k2??1??2??k3??2??3????kn??n?1??n??0??k1?k2??1??k2?k3??2????kn?1?kn??n?1??kn?k1??n?0??1,?2,??n線性無關,??k1?k2???k2?k3?????kn?1?kn???kn?k1??0?k1??k2?k3??k4????kn?k1,當n為偶數(shù)時,令k1??k2?k3??k4????kn?1,則?1??2??3??4????n?1??n?0也成立,這與條件不符.當n?為奇數(shù)時,則n?1為偶數(shù),則有k1??k2?k3??k4????kn?1?kn??kn?k1，立得k1?k2?k3?k4???kn?1?kn?0,等式k1?1?k2?2???kn?n?0才成立，這與條件完全相符.證畢.浙04184# 線性代數(shù)（經(jīng)管類）試題

第三篇：2013年10月自考線性代數(shù)真題

2013年10月自考線性代數(shù)真題

說明：在本卷中，A表示矩陣A的轉置矩陣，A表示矩陣A的伴隨矩陣，E是單位矩陣，|A|表示方陣A的行列式，r(A)表示矩陣A的秩。T

*

選擇題部分

注意事項：

1．答題前，考生務必將自己的考試課程名稱、姓名、準考證號用黑色字跡的簽字筆或鋼筆填寫在答題紙規(guī)定的位置上。

2．每小題選出答案后，用2B鉛筆把答題紙上對應題目的答案標號涂黑。如需改動，用橡皮擦干凈后，再選涂其他答案標號。不能答在試題卷上。

一、單項選擇題（本大題共5小題，每小題2分，共10分）

在每小題列出的四個備選項中只有一個是符合題目要求的，請將其選出并將“答題紙”的相應代碼涂黑。錯涂、多涂或未涂均無分。1．設行列式a1a2b1b2?1，a1a2c1c2??2，則

a1a2b1?c1b2?c2?

A.－3 C.1 2．設4階矩陣A的元素均為3，則r(A)= A.1 C.3 3．設A為2階可逆矩陣，若A?1B.－1 D.3 B.2 D.4 ??1?3??

?2?5???5?3?C.?? ??21?A.??13?*??，則A= ??25??1B.??2??5D.??23?? 5?3?? ?1?4．設A為m×n矩陣，A的秩為r，則 A.r=m時，Ax=0必有非零解 C.r

222B.r=n時，Ax=0必有非零解 D.r

5．二次型f(xl，x2,x3)=x1?2x2?3x3?8x1x3?12x2x3的矩陣為

?1?A.0???8??1?C.0???4? 0212026?8??12? 3???4??6? 3???1?B.0??0??1?D.?4??0?0?8??212? 03???40??26? 63??非選擇題部分

注意事項：

用黑色字跡的簽字筆或鋼筆將答案寫在答題紙上，不能答在試題卷上。

二、填空題（本大題共10小題，每小題2分，共20分）6．設A為3階矩陣，且|A|=2，則|2A|=______．

?12?7．設A為2階矩陣，將A的第1行加到第2行得到B，若B=??，則A=______.34??a12??a11a12??a118．設矩陣A=??，B=??，且r(A)=1，則r（B）=______.aaa?aa?a?2122??11211222?9．設向量α=(1，0，1)，β=(3，5，1)，則β－2α=________．

T10．設向量α=(3，－4)，則α的長度||α||=______．

TT11．若向量αl=(1，k)，α2=(－1,1)線性無關，則數(shù)k的取值必滿足______.12．齊次線性方程組xl+x2+x3=0的基礎解系中所含解向量的個數(shù)為______．

T

T?122???100?????13．已知矩陣A=212與對角矩陣D=0?10相似，則數(shù)a=______ ?????221??00a?????14．設3階矩陣A的特征值為－1，0，2，則|A|=______．

15．已知二次型f(x1,x2,x3)=x1?x2?tx3正定，則實數(shù)t的取值范圍是______．

三、計算題（本大題共7小題，每小題9分，共63分）

222a?b?c16．計算行列式D=

2ab?a?c2c2a2bc?a?bT

T2b2c.17．已知向量α=（1，2，k），β=?1，?，且βα=3，A=αβ，求(1)數(shù)k的值；

10(2)A．

?11??23??123??12?????18．已知矩陣A=231，B=00，求矩陣X，使得AX=B.?????340???10?????19．求向量組α1=(1,0,2,0), α2=(－1,－1,－2,0), α3=(－3,4,－4,l), α4=（－6,14,T－6,3）的秩和一個極大線性無關組，并將向量組中的其余向量由該極大線性無關組線性表出．

T

T

T?2x?3y?z?0?20．設線性方程組?2x?y?z?1，問：

?x?y??z?1?(1)λ取何值時，方程組無解？

(2)λ取何值時，方程組有解？此時求出方程組的解．

?001???21．求矩陣A=010的全部特征值與特征向量． ???100???2222．用配方法化二次型f(x1,x2,x3)=2x1?2x2?4x1x3?8x2x3為標準形，并寫出所用的可逆線性變換．

四、證明題（本題7分）

23．設向量組α1，α2線性無關，且β=clα1+c2α2，證明：當cl+c2≠1時，向量組β－α1,β－α2線性無關．

第四篇：線性代數(shù)02198自考2009年~2012年真題試題及答案(新)

2009年7月高等教育自學考試全國統(tǒng)一命題考試

線性代數(shù)試題

課程代碼：02198 試卷說明：在本卷中，AT表示矩陣A的轉置矩陣；A*表示A的伴隨矩陣；R（A）表示矩陣A的秩；|A|表示A的行列式；E表示單位矩陣。

一、單項選擇題（本大題共10小題，每小題2分，共20分）

在每小題列出的四個備選項中只有一個是符合題目要求的，請將其代碼填寫在題后的括號內。錯選、多選或未選均無分。

1．設A，B，C為同階方陣，下面矩陣的運算中不成立的是（）．．．A．（A+B）T=AT+BT C．A（B+C）=BA+CA a11a122．已知a21a22a31a32a132a11a23=3，那么a21a33?2a312a12a22?2a32B．|AB|=|A||B| D．（AB）T=BTAT 2a13a23=（）?2a33A．-24 C．-6

B．-12 D．12 3．若矩陣A可逆，則下列等式成立的是（）

1*A．A=A B．|A|=0 |A|C．（A2）-1=（A-1）2

D．（3A）-1=3A-1

?41?31?2??4．若A=，B=??23?，C=?02?1?，則下列矩陣運算的結果為3×2的矩???152???3?12????21??陣的是（）A．ABC C．CBA

B．ACTBT D．CTBTAT

3線性無關，則（α2,α3,α4，其中α1，α2，α5．設有向量組A：α1,A．α1，α3線性無關

4線性相關）

B．α1，α2，α3，αD．α2，α3，α4線性無關

C．α1，α2，α3，α4線性無關

6．若四階方陣的秩為3，則（）A．A為可逆陣

C．齊次方程組Ax=0只有零解

B．齊次方程組Ax=0有非零解 D．非齊次方程組Ax=b必有解

0???207．已知方陣A與對角陣B=?0?20?相似，則A2=（）

?0?2??0?A．-64E B．-E

C．4E

8．下列矩陣是正交矩陣的是（）0??10A．?0?10?

??00?1??D．64E

101?1??110? B．

?2??011?cos?C．???sin???sin?? cos???????D．?????2202216661063??3?3??? 3?3??3??9．二次型f=xTAx(A為實對稱陣)正定的充要條件是（）A．A可逆

C．A的特征值之和大于0

B．|A|>0

D．A的特征值全部大于0

0??k010．設矩陣A=?0k?2?正定，則（）

??0?24??A．k＞0 C．k＞1

B．k≥0 D．k≥1

二、填空題（本大題共10小題，每小題2分，共20分）

請在每小題的空格中填上正確答案。錯填、不填均無分。11．設A=（1，3，-1），B=（2，1），則ATB=__________.21012．若131=0，則k=__________.k2113．若ad≠bc，A=?ab?，則A-1=__________.??cd??14．已知A2-2A-8E=0，則（A+E）-1=__________.15．向量組α1=（1，1，0，2），α2=（1，0，1，0），α3=（0，1，-1，2）的秩為__________.16．兩個向量α=（a,1,-1）和β=（b,-2,2）線性相關的充要條件是__________.?x1?x2?017．方程組?的基礎解系為__________.x?x?03?218．向量α=（3，2，t,1）β=(t,-1,2,1)正交，則t=__________.19．若矩陣A=?10?與矩陣B=?3b?相似，則x=__________.???04???ax??22220．二次型f(x1,x2,x3)=x1?2x2?3x3?x1x2?3x1x3對應的對稱矩陣是__________.三、計算題（本大題共6小題，每小題9分，共54分）

11121．計算三階行列式124.141622．已知A=?23?，B=??3?1?，C=?0?11?，D=?120?，矩陣X滿足方程?????10????21???120???101??AX+BX=D-C，求X.23．設向量組為α1=（2，0，-1，3）

α2=（3，-2，1，-1）α3=（-5，6，-5，9）α4=（4，-4，3，-5）

求向量組的秩，并給出一個最大線性無關組.24．求λ取何值時，齊次方程組

?(??4)x1?3x2?0? ?4x1?x3?0??5x??x?x?0123?有非零解？并在有非零解時求出方程組的結構式通解.?1?6?3?25．設矩陣A=?0?5?3?，求矩陣A的全部特征值和特征向量.?4??06?22226．用正交變換化二次型f(x1,x2,x3)=4x1?3x2?3x3?2x2x3為標準形，并求所用的正交矩陣P.四、證明題（本大題共1小題，6分）

27．若n階方陣A的各列元素之和均為2，證明n維向量x=(1,1,…,1)T為AT的特征向量，并且相應的特征值為2.

2010年10月高等教育自學考試全國統(tǒng)一命題考試

線性代數(shù)試題

課程代碼：02198 說明：在本卷中，AT表示矩陣A的轉置矩陣，A*表示矩陣A的伴隨矩陣，E是單位矩 陣，|A|表示方陣A的行列式，r（A)表示矩陣A的秩.一、單項選擇題(本大題共10小題，每小題2分，共20分)在每小題列出的四個備選項中只有一個是符合題目要求的，請將其代碼填寫在題后的括號內。錯選、多選或未選均無分。1．設矩陣A=??1???1??，B=（1，1）則AB=（）

??A．0 B．（1，-1）C．??1???1???

D．??11?????1?1???

2．設A為3階矩陣，|A|=1，則|-2AT|=（）

A．-8 B．-2 C．2 D．8 abc?aabc3．設行列式D1=a1b1c1?a1，D2=a1b1c1，則D1=（a2b2c2?a2a2b2c2A．0 B．D2 C．2D2

D．3D2

*?

?12?4．設矩陣A的伴隨矩陣A??34?

??，則A-1=（）

A．?1?4?3B．?1?12?2????21?? 2??????34?? C．?1?2?12????34?? D．?1?2?42???31??? 5．設A,B均為n階可逆矩陣，則必有（）

A．A+B可逆 B．AB可逆 C．A-B可逆 D．AB+ BA可逆?103?6．設A為3階矩陣且r(A)=2，B=??010??，則r(AB)=（??001??A．0 B．1 C．2 D．3 7．設向量組α1=（1，2），α2=（0，2），β=（4，2），則（）））

A．αB．βC．βD．β線性無關

不能由α1，α2線性表示

可由α1，α2線性表示，但表示法不惟一 可由α1，α2線性表示，且表示法惟一 1，α2，β

?2x1?x2?x3?0?8．設齊次線性方程組?x1?x2?x3?0有非零解，則?為（）

??x?x?x?03?12A．-1 B．0 C．1 D．2 9．設A為3階實對稱矩陣，A的全部特征值為0，1，1，則齊次線性方程組(E-A)x=0的基礎解系所含解向量的個數(shù)為（）A．0 B．1 C．2 D．3 10．二次型f(x1,x2,x3)=x12+x22+4x32-2tx2x3正定，則t滿足（）A．-44

二、填空題（本大題共10小題，每小題2分，共20分）

請在每小題的空格中填上正確答案。錯填、不填均無分。11．行列式0112的值為_________.?12?12．已知A=??23??，則|A|中 ?12?19．與矩陣A=??03??相似的對角矩陣為_________.??20．二次型f(x1，x2，x3)=2x1x2+2x1x3的秩為_________.三、計算題（本大題共6小題，每小題9分，共54分）0121．求行列式D=201012210102的值.100?10?1?2022．設矩陣A=100，B=2?10，求滿足矩陣方程XA-B=2E的矩陣X.00100023．設向量組α1=(1,3,0,5)T，α2=(1,2,1,4)T，α3=(1,1,2,3)T，α4=(1,0,3,k)T，確定k的值，使向量組α1，α2，α3，α4的秩為2，并求該向量組的一個極大線性無關組.x1?2x2?x3?124．當數(shù)a為何值時，線性方程組?2x1?x2?x3?1有無窮多解？并求出其通解.（要求用它

x1?x2?ax3??2的一個特解和導出組的基礎解系表示）

25．已知3階矩陣A的特征值為-1，1，2，設B=A2+2A-E，求（1）矩陣A的行列式及A的秩.（2）矩陣B的特征值及與B相似的對角矩陣.?x1?2y1?2y2?y3?26．求二次型f(x1,x2,x3)=-4x1x2+2x1x3+2x2x3經(jīng)可逆線性變換?x2?2y1?2y2?y3所得的標準形.?x?2y3?

3四、證明題（本題6分）

27．已知n階矩陣A，B滿足A2=A，B2=B及(A-B)2=A+B，證明AB=0.

第五篇：2011年4月自考英語二試題真題及答案

2011年4月高等教育自學考試全國統(tǒng)一命題考試

Ⅰ.Vocabulary and Structure(10points 1 point each)從下列各句四個選項中選出一個最佳答案，并在答題紙上將相應的字母涂黑。1.You may use bike________ you give it back to me tomorrow morning.A.unless

B.provided C.though D.because 2.He could not ________his tears on hearing that he was not admitted to the university A.hold up B.hold back C.get over D.get through 3.Good parents have the ability to communicate messages of love, trust, and self-worth ______their children A.on B.for

C.with

D.to 4.The teacher required that all errors should be _______eliminated before the students turn in their term paper.A.deported B.eliminated C.deprived D.implemented 5.A person is lucky if his career ________with his interest and hobby.A.concerns B.competes C.coinciders D.compares 6.Out sleep influences our mood.our mood, ________,affects our performance.A.in return B.in vain C.in short D.in turn 7.It was ________of you not to disturb us while we were sleeping.A.considerate B.considering C.considerable D.considered 8.I'd appreciate it very much if you could make some ________on my recent article at the conference.A.requests B.references C.remarks D.restrictions 9.He is ______absorbed in his own thoughts that he seems unaware of what's going on in the room.A.so

B.each

C.very

D.much 10.No sooner(選C)had I reached home than Michael arrived with Jane in his car.A.did I reach

B.I had reached

C.had I reached

D.I reached

Ⅱ.Cloze Test(10point,1 point each)下列短文中有十個空白，每個空白有四個選項，根據(jù)上下文要求選出最佳答案。并在答題紙上將相應的字母涂黑。

It is difficult to escape the influence of television.If you fit the statistical messages, by the age of 20 you will have been esposed___11___at least 20,000 hours of television.You can add 10,000 hours for each decade you have lived ___12___the age of 20.The only things Americans do more than them watch the television are work an sleep.Calculate for a moment what could be done with even a part of those ____13____.Five thousand, I am sold, are what a typical college undergraduate____14____working in a bachelor’s degree.In 10,000 hours you corded have learned several lauguages fluently, you could be reading Shakespeare in the ____15____,and you could have walked around the world the world and ____16____a book about these hours.The trouble with television is____17____it discourages concentrations.Almost anything interesting and rewarding in life____18____some constructive effort.The dultest , the least gifted of us can achieve things that seem remarkable to those____19____never concertrate on anything.But television encourages us not to make any___20___.It makes the time pass without gain.11.A.to

B.under 12.A.at

B.from 13.A.decades B.hours

14.A.spends B.casts 16.A.written B.write 17.A.what B.that

18.A.requests B.asks

19.A.what B.whose 20.A.money B.time

C.for

C.after

C.years C.takes C.wrote C.which C.orders C.which C.effort

D.at D.before D.things D.uses D.original D.writing D.why D.requires D.who D.skill 15.A.origin B.beginning C.source

Ⅲ.Reading Comprehension(30point,2points each)從下列每篇短文的問題后所給的四個選項中選出一個最佳答案，并在答題紙上將相應的字母涂黑。

Passage one Questions 21 to 25 are based on the following passage.Michael Stadtlander set Toronto’s restarant world on fire in the 1980s with his original food, and eastablished a national and international fame.In 1994,however,he decided to leave the city restaurant scene and seek a quiet life in the country, where he has been preparing meals on his farm two hours borth of Toronto and asks guests to bring their own wine.Not long ago, Stadlander was charged with selling liquer without a licence, He said when a guest asked for wine at a dinner in December, he provided two bottles from his private stock.But the customer turned out to be an undercover Ontario Provincial Police(OPP)officer who returned a few days later along with four armed officers to seize 83 bottles of wine,and to search the farmhouse for records.The punishment could run as high as $100,000 in fines and a year in jail.When Mr and Mrs Smisth arrived at the farm on a weekend befre Christmas, they said that it was their annual celebrstion and asked if they could get some wine.Nobuyo,Stadtlander’s wife,told them the policy was thar guests peovide their own..Given the occasion, though, Stadtlander agreed to provide a bottle of white wiee from own cellar,at cost,as a favor, Later, Mr Smith asked for a second bottle and a detailed receipt that included the price of thr wine.He turned out to be OPP Detective Paul Smith and ―his wife,‖an assistant who went as a witness.―I was shocked,‖Nobuyo Stadtlander recalls.‖How could thy do this to us when we did them a favor?We sold them our wine at our cost.‖.But Mr.Smith say,‖ No.they made $20.‖

Stadtlander is determined to prove his innocence and continue the business.―People who live in the area have been very supportive,‖ he says.‖And when this is over,I want my wine back.‖

21.Stadtlander was accused because________.A.be offered too much wine for a guest B.be asked guests to bring their own wine C.be charged too much for the wine offered D.be sold wise to his guests without a license 22.In the 1980s,Stadtlander________.A.threw his original restaurant menu into the fine B.made his restaurant well-known around the world C.sought for a quiet life in the central part of the city D.opened one resyaurant after another home and abroad.23.IT turned out that Mr.and Mrs.Smith were________.A.two farmers B.two witnesecs C.hasbend and wife D.police officers 24.Nobuy Studtlander claimed that in the wine case they made________.A.$1.20 B.no money C.$100,000 D.a lot of money 25.Which of the following could be used as evidence against Stadtlander? A.What Stadtlander’s wife said B.The second bottle of wine

C.The records from the farmhouse.D.A receipt with the wine price.Passage Two Questions26 to 30 are based on the following passage.A recent study shows that sixteen out of every 100 American couoles have violent confrontations of one sort or another during the course of a year.In six of these cases there is severe kicking, biting, punching or hitting with objects.Almost four of every 100 wives are seriously beaten by their husbands.three of every 100 children are kicked or punched by their parents.More than a third of all brothers and sisters severely attack each other.As expected, the incidence of violence is highest among the urban poor(many of them minoroties).blue-collar workers, people under 20 or without religious beliefs, families with a husband who is jobless and those with four to six children.But the study also showed that violence occurs among wealthy families as well.Indeed, the wife of a university president once quietly called Straus, one of the sociologists who conducted the study, to ask what she could do about her husband, who often beat her.Straus suggested seeking assistance from marriage advisors.Steaus and his colleagues found out that there are various root causes that give rise to such behavior.―The reason are mixed –psychological, sociological, situational,‖ says Straus.―The husband, for example, may feel under particular stress because be has been out of work too long.Violence may also be an echo of the past,‖ Straus explains.―When Mummy gives her two-year-old a slap（巴掌）for putting something dirty in his mouth, he is learning from infancy that those who love you hit you.‖ Another reason may be the worsening economic situations.―If we have a real economos decline, It’s going to get worse,‖ economic situations.―If we have a real economic declines,it’s going to get worse,‖ says Gelles, one of Straus‖ colleagues.These sociologists have no easy answer to violence in the American family.While they welcome such move as the opening of shelters for beaten wives and the establishment of a National Center for Child Abuse and Neglect, they belive that there must be more basic attack on vilolence,including the reduction of ―macho‖（大男子主義）themes on the television,the outlawing of physical punishment in schools and perhaps even the wlimination of death senernces.As Straus explains,―Volence is an acceptable solution to problems in American society,And that is how it is used in families.26.Which of the following statements is true? A.More than a third of brothers attack sisters in American families.B.Almost four percent of husbands are seriously beaten by their wives.C.Six out of one hundred couples experience severe domestic violence.D.Six percent of American couples have some kind of domestic violence.27.The example concerning a university president’s wife shows that_________ A.domestic violence is found in well-to do families as well B.the incidence of violence is highest among urban families C.marriage advisons’ assistance is the best solution to violence D.domestic vilence exists regardless of age, race,and social status.28.What Gelles says shows that________.A.vilence is responsible for the decline of economy B.violence may be a reflection of one’s past experience C.violence is related to the economic situation of the time D.violence is the best form of emotional release for a husband 29.The word ―outlawing‖ in line5 of the last paragraph means________.A.making something illegal B.freeing someone from prison C.learning something from law D.throwing someone out of court 30.According to the sociologists, it is impossible to curb domestic violence unless________.A.more shelters for beateb wives are opened B.children are protected by a National Center C.violence is not accepted as a solution to problems D.Americans get tougher with violent behaviors Passage Three Qestions 31to35are based on the following passage.American scientists are developing an ―intelligent‖ mabile phone capable of blocking incoming calls depending on the owner’s mood., Using ―context aware‖

Technology the ―Sensay‖ phone will monitor calls and send back polite messages saying the user may be contacted later.A research team at the Institute for Complex Engineering Systems at Carragie Mellon University in Pittsburg, Pennsylvania, are developing body temperature and electrical skin monitors to help the device understand the emotional state of its user.If the phone senses that the user is busy —for instance, involved in a conversation—it might block an incoming call and turn it onto voicemail.The phone would send back a text message saying the user is unavailable, but advising that id the matter is urgent the caller can try again in three minutes.If a call from the same person came in again, the phone would put it through.The researchers are interested in four basic different states—busy and not to be interrupted, physically active, idle,and ―normal.‖ Most people are said to change between these states,an average of 6 to 12 times a day.Professor Asim Smailagic, a leading member of the Carnegic Mellon team, told The Engineer magazine.‖Today’s computers are pretty dumb compared with the device.We got to work at the beginning of May and since then have been improving it.The next stage is to make it smarter, adding various intellingence systems so it can learn about the user.The phone also employs four primary sensons—two microphones to pick up conversations and monitor local noise.a light detector and an acceleromster（加速度計）.The light sensor shows if the phone is being carried in a bag or pocket ,while the accelerometer determines whether the user is walking,running or standing still, In the sensor box, phone is being carried in a bag or pocket, while the accelerometer determites whether the user is walking,running or standing still,In the future,the sensor box,phone and personal organizer will be combined into one device.‖

31.According to the passage, the ―Sensay‖ phone is capable of ________.A.showing body temperature B.blocking unwanted calls C.detecting the owner’s mood D.sensing the caller’s mood

32.The second paragraph mainly tells us ________.A.where the intelligent phone is developed B.how the new phone is going to function C.whether the researchers failed in their experiment D.why the researchers failed in their experiment 33.Accorfing to Professor Smailagic, the detector will show________.A.where the mobile phone is B.where the mobile phone user is C.whether the mobile phone user is busy D.whether the mobile phone is within reach 34.the word ―it‖ in line 5 of paragraph 4 refers to________.A.the phone B.the monitor C.the computer D.the light sensor 35.The best title for this passage is ________.A.How to Tell a Person’s Mood B.How to Block Incoming Calls C.Sensay Your Personal Organizer D.Sensay The Futuer Mobile Phone

Ⅳ.Word Spelling(10points for two words)將下列漢語單詞譯成英語，每個單詞的詞類、首字母及字母數(shù)目均已給出，請將完整的單詞寫在答題紙上。

36.樂意地，容易地adv.r________ 37.前景n.p________ 38.永恒的,不斷的a.________ 39.有益的，有好處的a.b________ 40.進口n.i________ 41.預報n.f________ 42.元素，成分n.e________ 43.縮短，減少n.s________ 44.誠實n.h________ 45.有規(guī)律的a.r________ 46.改變，使不同v.v________ 47.坦率地adv.F________ 48.減輕，救濟n.r________ 49.和諧，融洽n.h________ 50.賠償v.c________ 51.有效率的a.________ 52.真誠的a.s________ 53.建設v.e________ 54.市長n.m________ 55.好奇心n.e________

Ⅴ.Word Form(10point each)將括號內的各詞變?yōu)檫m當?shù)男问教钊肟瞻祝鸢笇懺诖痤}紙上。

56.It will be interesting to see what’s going to happen at the next________(elect).57.For those involved, the scandal has been a very________(pain)experience.58.________(safe)needs to be improved on the railways in that country.59.The present crisis bears some ________(similar)to the oil erisis of the 1970s.60.His mother becomes disappointed whenever he fails to live up to her________(expect).61.Being a ________(high)motivated lauguage learner, he takes every opportunity to improve his English.62.Our most ________(success)product is based on a very simple idea.63.local people are questioning the ________(wise)of spending so much money on the new road.64.Don’t worry.This is the ________(desire)weight range for you health.65.This scientist was awarded with several titles for his________(accomplish)in biology.Ⅵ.Translation from Chinese into English(15points,3points each)將下列各句譯成英語并將答案寫在答題紙上。66.盡管有困難，他們最終還是把工作完成了。67.這次會議在促進兩國友誼方面起了重要作用。68.他長久以來夢想著稱為一名足球運動員。

69.在過去的二十年中，用于科學實驗的動物數(shù)量大大減少了。70.這次英語考試比我們預想的要容易得多。

Ⅶ.Translation from English into Chinese(15points)將下列短文譯成漢語并將答案寫在答題紙上。

Shy people are likely to be passive and easily influenced by other.They are very sensitive t criticism.They also find it difficult to be pleased by praises because they believe they are unworthy of them.A shy person may respond to a praise with a statement like this:―You’re saying that to make me feel good.I know it’s not me.‖It is clear that, while self-awareness is a good quality.overdoing it is harmful.Can shyness be completely gotten rid of ,or at least reduced? Fortunately ,people can overcome shyness with determined and patient effort in building self-confidence.