居余马线性代数第三章课后习题

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.=>(A8)X=06X=0B2CDᑣ"AB)=r(B).>(A8)X=0BX=0B2CDᡠI(AB)X=0ḄJẠLᡠMᔣ
ḄᦪBX=0ḄJẠLᡠMᔣ
ḄᦪOP.ᓽ=஻"3),4,r(AB)=r(B).33.7AQx஻;▣8nxs;▣=>AB=0,ᑣr(A)+r(B)W஻.78=(4,…,U)ᐸ-V…UDᑡᔣ
ᵫA8=0XA/7,=0,/=1,…,s.>r(A)=r,ᑣAX=0ḄJẠLMᨵ஻[ឋ]ᐵḄᔣ
PᡝI9.…e>AX=0Ḅᔣ
ᑣ/…,&`ᵫAX=0ḄJẠL[ឋ⊤bᡠIr(S)

20or(A)=n—lqᓽxyᨵ஻1▤z{|Pl}ᡠIA*HO,~|A|=0,A*wO,ᡠI/"(A*)21.=0,ᡠIA4*=0,ᓽA*Ḅᑡᙳ[ឋ2CDAx=0ḄᡠIr(A*)2),=(A*)*=M2A.A஻▤`⌮;▣ᡠIA*஻▤`⌮;▣~pr|=|AA*(Aj=„,ᡠI(A*)*=k*|(A*)T஺A44*=|n/,ᡠI(A)ᑄ஺4,36.74஻▤;▣=[ឋ2CDAX=bbᨵḄ

21ᐙ⌕ᩩWko.ᐙᑖឋB¡0,ᡠIr(A)=஻=r(A,b)஺4l஻r(A)=n=r(A,b),4X=bᨵ./ஹ%£⌕ឋ(¤¥)Ꮇ7|n=0,ᑣ(4)<஻஺7A=aᑣa”[ឋOᐵ¨ᐸ-xyᨵᔣ
©ᵫᐸ"ᔣ
[ឋ⊤ª|«7a“`ᵫ/,஺2,…[ឋ⊤ª'%I0]¬b=(0,0,…,0,1)7,|J(A¡)'H,ᓽr(A)<«A,b),ᡠI2CD]0I஺U³´஺37.7X]-,¶2¶3==%/=¶5·¸%5=¹2CDᨵḄᐙ⌕ᩩZq=0,ᙠᨵḄ½¾¿ÀªÁḄÂ஺/=1/-x=%,x-x=a,x-x=a,x-x=a,x-x=a,22323434545(5ᨵ

221-1000q01-100a2001-10a3->----->0001-1ஹ—10001ஹq-1000%01-100a2001-100001-1«40000q+ᓡ+஺3+஺4+஺5,ஹr1000-P'1000-1-11000-11000஺4ÅA=0-1100,ÆÇ;▣(4¡)=0-1100஻300-11000-110a2ஹ000-100-1152CDᨵḄᐙ⌕ᩩr(A)=r(4¡)ᓽÃÄ=0஺1=1o=°q/=1ஹ'1-1000%1"1000-1a+஺2+஺3+஺4i01-100஺20100-1ᓝḄ+஺4001-10%f----->0010-1+%0001-10001-1஺400000>ஹ000000)¬ZÈᵫÉ
Åx5=0X2CDḄᱯX஺=(4+a+a+a,a+a+a,a+a,a,0)7v23A234344·¬Ë=1Xᐸ̪DḄJẠLX,=(1,14,1,1)

23q+Ḅ+%+a4a+a+%23ᡠI2CDḄÂX=Xo+AX|=4+4+᝕1ᐸ-ÎÏᦪ஺38.Ð42CDAX=஻ḄÑ|B%,%Ò[ឋ2CDAX=0ḄJẠLᑣAX=bÂ(A)kiai+k2(a]+a2)+—~;kxax+Î2(%-«,)+(C)A]%+k.Bஹ+Ô)+'[42vÕᓃ%+k2(᜛1—)+'2•:`X%%-V[ឋ]ᐵḄ~AX=0Ḅ4AX=0ḄJẠLÙ⎃AX=6Ḅ4⌱(B).2’123ஹ39.Ð஺=24t};▣PQ=0,ᑣஹ369,(A)o"6ᔢr(P)=l;(B)o=6âr(P)=2;(C)oqr(P)=lv(D)oqr(P)=2v[123]PQ=0,~஺=24ᡠI“P)+r(Q)W3,P};▣ᡠஹ369,Ir(P)Nl,or*6qr(Q)=2,414“P)<1,ᓽr(P)=l,ᦑ⌱(C).40.7%=(%,%,஺3)'a2=(å)'a3=(C,,c,c)r,ᑣÃᩩç[23+=0,(è2+”2w0),«=1,2,3)élêḄᐙ⌕ᩩ:

24(A)஺ë,஺2஺3[ឋOᐵ(B)[ឋ]ᐵ;(0r[a,a,a}-r{a,a}(D)%,a2ª[ឋOᐵ%,%[ឋ]ᐵ.]23t2aix+biy=-C[(qh]—Jஹ:=QᨵìḄᐙ⌕ᩩr஺2b2rab-c=2,222+63y=-C/3%-C3>3%,/>ஹabஹ-C)}rab-c=2,ᓽ%,%஺3[ឋOᐵ஺222/34-C3>åஹra%=2,ᓽa”î[ឋ]ᐵ஺ᡠI⌱(D)஺2\a383,41.7Amx஻;▣r(A)=ïQð஻)8஻▤;▣¿ᑡñᡂ?(A)A-ó▤z{=0;(B)A-஻?ᑡ[ឋ]ᐵv(0u74#0;(D)>A5=0,ᑣ8=0v(E)>r(B)=஻,ᑣr(AB)=m.⌱(E).r(B)=n,ᡠI8`⌮r(AB)=r(A)=m.42.7ᑄ஺?,…Ë“(%eR",i=1,…óó2)[ឋ]ᐵ¿ᑡñᡂô(A)Ïᦪᓃõ&….&ᨵ஽Ë+᝕2a2+…+ᓃ=0v(B)Î÷ீùᔣ
Ë…[ឋOᐵv(0peR",a,,...,a,„,B[ឋOᐵ;

25(D)WXk(k<஻?)ᔣ
Ë…[ឋ]ᐵ.⌱(D),᦮û[ឋ]ᐵüᑖ£[ឋ]ᐵ஺43.7a,4,7[ឋ]ᐵ/᜛»[ឋOᐵ¿ᑡñᡂô(A)a£`ᵫ[ឋ⊤bv(B)4£`ᵫa,7»[ឋ⊤bv(0b£`ᵫa,£,y[ឋ⊤bv(D)b£|`ᵫa,[ឋ⊤b.⌱(C)஺a,/?,y[ឋ]ᐵᡠIa,£[ឋ]ᐵ஺a,᜛[ឋ]ᐵa,p,8[ឋOᐵᡠIb£`ᵫḄ/[ឋ⊤bb£`ᵫa,7[ឋ⊤b஺44.7A4x3;▣"4)=1,èþ3[ឋ2CD4X=6ḄÃ[ឋ]ᐵ¿ᑡñAX=0ḄJẠLô(A)(B)஺+&—2᝕(C)-YZ4H3
r(A)=l,ᡠAX=0ḄẠ
ᨵ2ឋᐵḄ
(A),(B)!஺(D)Ḅ$
%AX=0Ḅ
ᦑ⌱(C).45.,ᔣ./01,஺2஺33ឋ4ᐵ0஺2஺3«.3ឋᐵ஺678ᑡ:⚪<=>?஺(1)%Bᔲᵫ0%E3ឋ⊤GH(2)(Z4Bᔲᵫ0஺1஺2஺3)ឋ⊤GH
(1)12஺3஺4ឋᐵᡠ124Jឋᐵ

26KL,%ឋ4ᐵᡠMNᵫ஺2஺3ឋ⊤G஺(2)(O=P)Ꮇ,%Bᵫឋ⊤GRS(1)%BSE஺3ឋ⊤Gᡠ஺4BS%஺3ឋ⊤Gᓽ4%%ឋ4ᐵU஺2஺304ឋᐵVW஺ᡠ«BS0ᑄ,4%3ឋ⊤G஺446.,A஻▤\▣^_ᙠ᦮ᦪk(kN2)efA%=0,gh-jk0(ᐸma஻opqᑡᔣ.)=>a,Aa,…hsjឋᐵ஺=>(tuP=)^.a+24a+…+"Aia=Oz{$|}~ftAk-'a+tAka+---+tA2k--a=^x2kA*a=O,ᡠ4ᓃ=s=A?"2a=0tAk-'a=0,K4"j/0,f.=0஺tᑭᵨ᪵PNf2=h-h=0…,"஻ឋᐵ஺47.,4,8ᑖ஻x஻?,஻?x஻\▣(n8Ḅᑡᔣ./ឋᐵ஺=A8=/,Un

27ᦑ8Ḅᑡᔣ./ឋᐵ஺48.{%஺2஺3}=᜛2,᜛3)ᐸm%=(1,2,-3)7,02=(3,0,1)7,%=(9,6,-7d=(0,1,-l)r,/?=(஺,2»,=("IQ),Nᵫ2L,஺?6^ឋ⊤G஺,bḄ¢஺‘139b](\-11-1'
£஺|,஺2஺3᜛3¥=206102430)(00ஹ-31-70-5+U3Nᵫ᜜&2Eឋ⊤Gᡠᨵ6—5=0,8=5஺'139)(139ஹ(%,%,%)=206ߟ012©31-V100ᡠ{%%,஺3}=2஺’0஺5ஹr-i10ஹ(ᱏ⊈᜛3)=121->-»01i1-11oj,0஺5-M{4,,/?3}={%%%}=2,ᡠ5-ga=0,ᡠa=15o’1a…aஹa1•a49.,A=....¬▤\▣(஻23),QER,r(A)=஻sl,஺஺aQ••1,
r(A)=஻s122(n>3)ᡠQW1

2800-1000-1000(n-l)a+lr(A)=”sl,ᡠ(஻sl)a+l=O,,50.,஻▤\▣AḄ®¯ᐗ±?²ᙳqKr(A)=/i-l,´µឋ¶/Ax=0Ḅ·
஺
r(4)=஻-1,ᡠ´µឋ¶/Ax=0ḄẠ
ms
ᔣ.஺,A=(4ᑖ…¸,)AḄ®¯ᐗ±?²ᙳqᡠ4+¹+…+ᵨ”=0ᓽA1=0,I%´µឋ¶/Ax=0Ḅ•Ạ
஺¼Ax=0Ḅ·
:&=k9ᐸm¾¿Àᦪ஺51.Á8ᑡឋ¶/I,n
ឋ¶/ÂᦪÃ஻/?¢஺X]+X)—2%4=—6,I<4XjsÈ2sk3sÉL3R-x-x=3;23%)+mx-x-x=-5,234II:Ìnx-x-2X=-l1,234

29'110-2-6ஹ’100-1-2ஹ
4-1-1-11—•,—010-1-4ஹ3-1-103)ஹ001-2s5,ᡠ(-2,-4,-5,0y%¶/IḄs
¶/IUII
ᡠÎJ%¶/IIḄs
ÑÎÒᐭ¶/II,Nfm=2,n=4,t=6.52.,a=(l,2,l)7',£=(l,,0)7'7=(0,0,8)T,A=a£6=//a,
¶2B2A2X-A4x+B4x+y஺
ᓽ
p´µឋ¶/—-840f10-11ஹ2QFI—BJAjyh16-800-...>01-218JI஺84-160007ᡠ(2B2A2-AA-B^X=YḄsᱯ
(,1,0Ý஺(1,2,1)ᐸÞß/ḄsẠ
஺1(Ll,0)r+Ml,2,l)r,ᐸmE¾¿Àᦪ஺QB??!?-A—6")x=yḄsá
:253.,n▤\▣A=(%,Ḅ…ã)Ḅ¯ᑡ{▂஺0AḄå“s1ᑡ᪀ᡂḄ“x(i—1)\▣é4%-1):¶/A]X=%ᨵ
ᔲHêëH

©(&)=”s1,“44,)=஻஺54.,a,£ᙳpqḄ“oᑡᔣ.A=a/3T,=>Am¾¿$¯(ᡈ$ᑡ)ᡂíî஺
r(A)«min(r(a),*£T))=l,ᡠAm¾¿$¯(ᡈ$ᑡ)ᡂíî஺

30AIA255.,஻▤\▣AᑖᙽA=9ᐸm}▤N⌮\▣(ᦇ<஻),=>_A?i^22>4oᙠòóôᐗ1Ḅzõô\▣U²8õô\▣L,efLAUஹ0B
ᵫᑖᙽ\▣Ḅö÷øᣚú⍝:40ஹAlA2(ikAi0M▂G42஺°42-A]?/n-k70’4—ü4ஹᡠL=9U—A2]A]]n-k10/J56.,A,BḆ஻▤\▣=>:1B..(1)Aᔆ”Ẇ(2)\l-AB\^\I-BA\-(3)det(AI-AB)=det(AI-BA)(2
ᦪ)஺OY1O]_//Bஹ:(1)“A/ᔆ[0I-ABஹA0IBIBᡠ/Al-0I-ABB,4-B\I(2)J=lAI,0/JUᡠ,-BIB\\I-BA0Z=A0IA

31B=/_ᕒ4A111ᵫ(1)ᓽ—஺বᑖ#$%&ᩭ()஺*4=0,-4.=(1)"01|=-ᡂ4஺*/IwO,,IoYI:_|7B\(I-7BYZ0ஹ஻ᔆ10AI-ABJ10I஻J(AR,Aᡠdet(2/-AB)=det(A/BA)=஺AAIFGH)ᡂ4஺57.KLAM஻zx஻P▣r(A)=r,ᑣTᙠ஻zxrP▣8,rx஻P▣C,Xr(8)="C)=r,YA=8C(Z[ᑭᵨ^_᪗ab)஺(I0ஹKr(A)=r,ᡠTᙠc⌮P▣e(f▤)ஹ஺(஻▤)Y24஺='

3258.A,BḆ஻▤P▣e4+*8஻KTᙠc⌮P▣஺YAQB=O஺KHᔠ^_᪗ab⍝Tᙠc⌮P▣ᱏ,Q”,஺2,YW4o'ojrA+rB஻ᡠ=0,{஺=஺Zᑣ⚪஺59.K4%,…,?ᐸ4*0tឋ^ᐵḄᐙ⌕ᩩMTᙠ¡¢1»¢eY6cᵫ¥,஺2…,%_]tឋ⊤[X⊤[¨©஺KᐙᑖឋTᙠ--¡ª1«WrYq.cᵫ%,஺2,…,tឋ⊤[ᡠ,tឋ^ᐵ,®…¯tឋ^ᐵ஺°⌕ឋ±,஺2,…,¥tឋ^ᐵᡠTᙠᐰ³Ḅsᦪ&&…ᓃ•Yk]%+kaH---1-ka-022rrᙠYᓃq+k2a2+…=0ᡂ4Ḅᡠᨵ³Ḅ»ᦪ°ᨵ¡ᨬ½Ḅ¾᪗Yk¿0,À%=0/ுi஺¾☢ÃKli«r஺᝞Çi=l,ᑣᓰ஺1=0É¿0,=0ÊË஺ᨬÌ⊤[¨©஺L,஺2,tឋ^ᐵᑣÍᯠᑮsᦪÐÑ☢ᓃḄÒ¨ÊË஺ᡠ:,஺?,…,a”|tឋvᐵ஺Ó±,஺2,…,a,tឋ^ᐵᡠ⊤[¨©஺60.Kᔣrsa”஺2,…,¢tឋvᐵḄᐙ⌕ᩩMa¿£Õ¿«=2,3,…,s஺ØτZ[Ù⚪M59⚪Ḅ⌮ᔲÙ⚪஺61.ᔣrsª,஺2,…,¥tឋvᐵ᝞ᙠᔣrsḄÑ☢Ûᐭ¡ᔣr£Kᙠᔣr

33sᨴ,ª,%Þßàᨵ¡ᔣr%(14,4e)cáᐸÑ☢Ḅi¡ᔣrâª,஺2,…%-1tឋ⊤[஺ãᙠR3äåæçè஺Kéᨵ#¡ᔣrᙳcáᐸÑ☢Ḅᔣrtឋ⊤[«,-kj3+ka-----F&_]/_](1)]]%=//7+/]/+••••(2)পx/—(2)x᝕(kJ-1/)%4-----F(&-]/_/”#)«_]—(/+lk)a-lka-------1+ka^=0iii+]i+],Þ…,¥tឋvᐵᡠ,஺2…•?tឋvᐵª,஺2,…¥tឋvᐵk=Q,ᑣᵫ(1)ªcᵫᑁ,&2,tឋ⊤ïÐ%,Þ…,¥tឋvᐵÊË஺62.Kᙠ஻ðᔣrñòR"Lᔣracáᔣrs%,4,…,4tឋ⊤[ᑣ⊤[¨©Ḅᐙᑖ°⌕ᩩMᔣrsaஹtឋvᐵ஺K(ᐙᑖឋ)ᨵ⊤[¨a=▾`+ka+…+ka22ssa-/,«|+la+•••+la22ss#ó^ô:(%-/])£]+(᝕2—2)42+…+(ᑍ—S)4=0ª,¥…,4tឋvᐵᡠᓃ=÷&=4,&=½ᓽc⊤[¨©஺(°⌕ឋ)é஺2,4tឋ^ᐵᑣTᙠᐰ³ḄsᦪP1P2,…,P,Y

34pa+pa+•■■+pa=0]i22ssᔣr஺cáᔣrsa1,4tឋ⊤[ᡠTᙠsᦪª,°…øYᡠ,a=(p|+4)%+(஻2+%)%+…+(Ps+%)&±“2,…,P,ᐰ³ᡠúMûüG☢Ḅý$⊤[¨Ð⊤[¨©ÊË஺63.AM”▤P▣,r(A)=loK:(1)A=(þÿ…);(2)A2=M.(1)r(A)=l,ᡠAḄᔣᡂᓽ!"#஺ஹ/ஹqaiaaপA22(4,…)2(4,…)()=(*…)?ᓽ!"#஺/a\\a\2.jn.a2\a226r4./AA=.:",3=J,஺=(஽…)r,X=(X],X2...,Xm)r.\aml.amol^mn7\y(i/(1):Ay=Z,ᨵ>ᑣ4%=0ḄBCD>ᓰ,32,…,/FGHIJ

35^+bx+-+bx0.22mmA'(2)IJDA)=bᨵ>Ḅᐙ⌕ᩩOPIJDR>(ᐸT0P"xlVW▣)஺bT)(1)Ay=bᡠ/!YBCDZM2,…,[9:\GHA'x=0,ᑣFᨵ]"ᨌ=0,ᓽZ/x=0,ᓽ6ᑁ+4/+…+%=0.(2)IJDAy=bᨵ>=r(A)=r(A/)=baᵫ4ḄᑡᔣDdឋ⊤ghA1A'0(F⌕ឋ)baᵫAḄᑡᔣDdឋ⊤gᡠr(A)=r஺“JIIfATY(AT()ஹ(ᐙᑖឋ)IJDTx=R>ᡠr(A7)4rw4=(%…J,i=xystdឋIJDZ/y)=°i=1,2,…஻C"Z)=1ḄuẠ>wᡠ|>ᔣḄpᦪ~ygCpuẠ>w஺

36>stdឋIJDZᓃ=0,i=l,2,...,n-mj=iḄuẠ>wᡠ|>ᔣḄpᦪ-q=஻z஺66./qx஻W▣AḄqpᔣPstdឋIJDCx=0ḄCpuẠ>w8PC,pq▤a⌮W▣஺BAḄᔣP=0ḄCpuẠ>w஺᝞…”3/4=7,B=??"஺ᑣᩩOCa=0i=l,2,…,1%b"i2…Mm/F஺2,…"dឋRᐵ஺•••*,Fba+ba+---+b,ail]l22mmBA=?%82wa221%+b22a2HC/d,+%02+…C[a+᝞e+…+fᡈ(=%Cah+%Ca+…+ᑗ/a“=஺ᡠBAḄᔣPCx=0Ḅ>஺8a⌮AḄpᔣdឋRᐵᡠBAḄqpᔣdឋRᐵ!BAḄᔣPCx=0ḄCpuẠ>w஺67.:A”▤W▣஻>1,ব=0,ᑣTBᡈᑡYᐗ¢Ḅ£ᦪ¤¥¦ᡂ஺:ভ=0,ᡠrA4"-l,!4*41,ᓽa஺68./AP஻-lx஻W▣⊤©ATᑜ«¬/ᑡᡠ᪀ᡂḄᑡ¦஺(1)(ߟ|AJ,|▏,…,(ߟ1)"I)7■PAx=0ḄCp>;

37(2):(j=l,2,...,n)²ᐰVᑣ(1)TḄ>PAx=0ḄCpuẠ>w஺(1)(a=(l,l,…,1),᪀⌼▤W▣8=(°],²µ⍝8T¬Cᐗ¢Ḅ£ᦪ¤¥¦ᑖ·ব¸▂1º»஺ᡠATḄᐗ¢¼(ভH▏,…(C1½ভyᙳ0,!4(C▂,▂...(C1)"▂)=C4(|▂,C|4|,...,(C1)”"»¿=0(2)(£=(0,0,…,0),᪀⌼஻▤W▣஺=",ᑣ²µ⍝CT¬Cᐗ¢Ḅ£ᦪ¤¥¦ᑖ·ভ,C%|,…,(C1º|Á஺|&|()=1,2,…,஻)²ᐰVᡠCḄÃÄW▣C*H0,ᓽr(C")Nl,!r(C)2"-1,Æᯠ—1,ᡠ«C)="—1,ᡠr(A)=஻-l,mnstdឋIJDAx=0ḄuẠ>wT|-r(A)=lp>ᔣ஺Zᵫ(1)Ê|Aj(j=\,2,...,n)²ᐰV!⚪஺69.:4Cp஻▤W▣&2=A,r(A)+r(A-/)=n.Æᯠr(A-I)=r(I-A).஻=r(7)="A+/-A),!—QPr(A)+r(A-/)

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