線性代數(shù)第二章

1、習(xí)題習(xí)題2 .32 ,512231,402753bababa求已知914984512231402753：ba解5122313402753232ba732813.,2,662479154257,864297510213zbzaba求且已知.16111122223286429751021366247915425721)(21：abz解., 0)(2)2()2( ;3) 1 (，101012121234,432112122121yybyababa求若求設(shè)；ba139732828513110101212123412963363663633) 1 ( ：解.22323234034022310310322

2、)2(bay.,2103,1231baabba求.25661231210321031231： baab解計(jì)算下列各式:.20413121013143110412)4(;00000010001001) 3( ;127075321134)2( ;123321) 1 (cbak10132231) 1 ( :原式解;00000)3(cbkba原式.6520876)4(原式;49635)2(原式.)(,2412,435214,240031tabccba求設(shè).4226140602412435214240031)( :tabc解.,)(,22baabbababanba證明：若階方陣均為證明:由題可知:222

3、2()()ab ababaabbab.abba所以已知線性變換,5232232133212311yyyxyyyxyyx323312211323zzzzzyzzy求從變量321,zzz321,xxx到的線性變換.321321321321310102013,511232102:zzzyyyyyyxxx解由此可得32132132116419412316310102013511232102zzzzzzxxx.164941236321332123211zzzxzzzxzzzx即.,10001001,00010001,333231232221131211caacbaab，kckbaaaaaaaaaa求設(shè),

4、.,：333231232221131211333231232221131211kakakaaaaaaabakaaakaaakaaaab解.,333231132312221121131211333232312322222113121211aaakakakaakaaaaacaaakaaaakaaaakaaac.)(4(;)(3( ;)(2();()(1 (1221,0121,1110ttttttababbababcaccbabcacab，cba驗(yàn)證已知).()(;662121451110122101211110)(;662112212201122101211110)(1 ( :bcacabbcac

5、ab所以證.)(;5457214533121221012112211110;545712211231122101211110)(2(bcaccbabcaccba所以.)(;13210211111001211110;1321123101211110)(3(ttttttttttbabababa所以.)(;20211110021111100121;2021220101211110)(4(ttttttttttabababab所以將下列矩陣化為行最簡形:12433023221453334311)3(174034301320)2(4311) 1 (10014311) 1 ( :解0000310050101

6、74034301320)2(.0000001000221003201112433023221453334311)3(221( )53,33f xxxa已知( ).f a求2( )53,21212110533333330100.00f xxx解:35)(2aaaf計(jì)算下列各式:20012(1);(2) 00.3400nabc212510(1).3415100000(2) 0000 .0000nnnnaabbcc解:.,001001kaa求設(shè)由二項(xiàng)式定理可得解,000100010100010001001001a：be nkkkkkkkkkbbkkbkebbekkbekebea2212212) 1(

7、)(2) 1()()()(.0002) 1( 0000001002) 1(000100010100010001222221kkkkkkka：kkkkn所以, 000000010000010001032nb，b，bb其中證明:(1) 兩個(gè)同階的上三角矩陣的乘積仍為上三角矩陣;(2) 對于任意矩陣a, aat和ata均為對稱矩陣; (3) 設(shè)a,b是同階對稱矩陣, 則ab也是對稱矩陣的充分必要條件是a與b可交換, 即ab=ba.)(3(.)()()()()2(.) 1 (：baababbaababa，aaaaaaaaa，aaaaaattttttttttttttttt均為對稱矩陣所以略證明用逆矩陣定

8、義求下列方陣的逆陣 .;13431) 1 (a.)( ,1012,0121)2(111abbaba求設(shè).1232)(;10201121;10112021)2( 1 ( :21214341112121*12121*1*1abbbbaaaaaa解;12643152) 1 (x.101201325120112)2(x用矩陣乘法 解以下矩陣方程:;8023212643152) 1 ( :1x解.381326101201325120112)2(1x.,323321321323312211的線性變換表示求用已知線性變換zzzyyyzzyzzyzzy.923192,3132,913

9、1913101020133213321232113211321yyyzyyyzyyyzyyyzzz所以,310102013:321321zzzyyy解.,) 3(; 00,)2(;,) 1 ( :1也是對稱陣則是對稱陣若可逆陣則且為同階方陣可逆若方陣則其逆陣惟一可逆若方陣證明aabab，baa.,)()(,) 3(; 000,)2(;,) 1 (:1111111也是對稱矩陣所以故為對稱矩陣存在則可逆所以則的逆矩陣同為設(shè)證明aaaaaaabaabaaacecbacbebecaeacbaabacbttt., 0)2(; 0, 0) 1 (:,1*naaaaaaa則若則若證明的伴隨矩陣為若階矩陣.

10、0.0; 00)(0)(, 0,)(, 0, 0; 0, 0, 0) 1 ( :*1*1*1 *aaaaaaaeaaaaaaaaa所以必有的假設(shè)矛盾與得由存在則設(shè)若從而則若證明.,:,)2(1*nnaaeaaaeaaaaa從而得等式兩邊同取行列式由定理知設(shè)方陣a滿足關(guān)系式 , 試證a及a+2e均可逆,求出逆陣. 0222eaa;2)2(,2)2(022:12eaaaeeaaeaa且可逆從而得由證明.64)2(,2,6)4)(2( :, 0226)4)(2( :12aeeaeaeeaeaeaaeeaea且可逆從而所以又因?yàn)?323432111)4( 602130321)3( 6432)2( 52

11、21) 1 (判別下列矩陣是否可逆，若可逆，求其逆矩陣.;6432, 06432)2(;12255221, 015221) 1 ( :1不可逆故且故矩陣可逆解., 0323432111)4(;431234102147329, 04602130321)3(*1故矩陣不可逆且故矩陣可逆aaa.,2001,1141,111apapp求其中設(shè).68468327322731 242124213120011141 ,2001,:111113133131343111313134311111111111111111apppppppappaapp故其中從而得由解個(gè)利用初等變換求下列矩陣的逆矩陣:;1210232

12、112201023) 1 (a.)2(,300341003)2(1ebb求;2210100:)()() 1 ( :56515253535152535111aaeea可得利用初等變換解變換初等行.100001)2()2(2321211eb同上可得;101311022141)2(x解矩陣方程;234311111012112) 1 (x32538122111012112234311) 1 ( :1x解;011110210132141)2(411x.2)3(,21,3,*1*的值求的伴隨矩陣為階方陣為已知aaaaaba.27161)32()32(222)3(,:3131321131*131*11*1a

13、aaaaaaaaaaaaaaa解.23)4( ;)3( ;2)2( ;) 1 (:. 3,3*1*21aaaaaaa求階方陣為. 92323)4(;93)3(;729822)2(;311, 1) 1 (:*121*3211aaaaaaaaaaaaaaan所以因?yàn)榻?3, 3, 2,5,1*的值求行列式階方陣均為設(shè)bababa.12961333:451*51*bababa解).0 , 0 , 0 , 1, 1 (),0 , 0 , 1 , 0 , 1 (,4使其中兩行為的矩陣求作一個(gè)秩為.0000001000001000001100101:所作的矩陣為解求下列矩陣的秩,并求一個(gè)最高階非零子式:0

14、2301085235703273812)2(443112112013) 1 (ba; 01113, 2)(.000056401211564056401211) 1 ( :的最高階非零子式為：解aara. 0001532712, 3)(00000140000712100230171210024205363002301)2(為的一個(gè)最高階非零子式bbrb; 11110421106312121011 )2( ; 512311124031 ) 1 (.0722134305143212031214013)4(0510317145024)3(求下列矩陣的秩. . 200007170403171707170

15、4031512311124031) 1 ( :2321131232rrrrrrrrr所以解; 3 000005300042110210111111042110421102101111110421106312121011)2(23432412rrrrrrrrr所以; 2,00000023012100023023012110150101504601210510317145024)3(2321251334122114135107rrrrrrrrrrrrrrr; 3,00000000001123310033810143213381000000000004895014321111140816181004

16、895048950143210722134305143212031214013)4(24232512311514132253rrrrrrrrrrrrrrrrrddccbbaa000000010001010001000000利用分塊矩陣計(jì)算222222220000000010001,0010001000000:deceeddccbeeaebbaa解.)(0100)(010000)(,)(002222222222222222bdcbdcacaacaebdceaceaeebdceaceaedeceebeeae:均可逆階矩陣及階矩陣設(shè)bsan；cobaoc1,) 1 (求若.,3221,503512101)2(1cba求時(shí)當(dāng).00000000:,00000000:,000000) 1 ( :1111111111nsnsnssnnsnsnssnnssnnsabeeabbaab