4ม7-8-64

เมทรกิ ซเ์ อกลักษณ์

เมทรกิ ซเ์ อกลกั ษณ์

ทิ ม  n I n = I jk nn

jk n

In I jk = 1 , j=k
0 , jk

เมทรกิ ซเ์ อกลกั ษณ์ (identity matrix) n  n

In I

1

I1 = 1 I1 = 1

I2 = 1 0 I2 = 1 0
0 1 0 1
0 0
1 0 0 = 1 1 0
I 3 = 0 1 0I 3 0 0 1
0 1 0
0

ก รคูณระหว่ งเมทรกิ ซ์

ก รคูณระหว่ งเมทริกซ์

ทิ ม  B = bij nr  AB = cij mr

 A = aij mn i 1,2,3,...,m

cij = ai1b1 j + ai2b2 j +  + ainbnj j 1,2,3,...,n

n n  n
   a11 a1k bkr 
a12  a1n b11 b12  b1r   k =1 a1k bk1 a1k bk 2 k =1 
 a22  b21  b2r   n n
a21   a 2n  b22   =  a2k bk1 k =1  a2 k bkr 
      k =1  n k =1 
 am2     bnr   a m k bk1
 bn1   n a2k bk 2 
 
k =1



n

amkbk 2

k =1
  am1 am n bn 2 n
 
 k =1 a m kbkr 

k =1

n

 aik bkj = ai1b1 j + ai2b2 j +  + ainbnj

k =1

2

1. 1.A23 AB2334 2. 2.B34 CD3112 3. 3.D12 EF2233 F23
C31 E23

ธ 1.ธ 1. AA B 3 =B 3 3 =

ธ EE223B323121B123..DB....4CADBFF2231331113=..==. 3C12A3ACAAA.ADBABDB2B2233C3 2B344 2  42. C31 D12 3. E23 F23
D3.
23..ธ B34 2.C C31 CA D12 3. E213=DB1 F2133=
3. ธ B3342 32224. C31 AD12
A32.ธ3 E23 B F233
D12
1. 31 2  4 3E. E2AC3 E F23 F BD 3F 231=
A
AB 2D3.. 12=14CD 2342 CB 3 = 3 D1
D F1=
32 C E

3. F C3D 2 C 3 2 DE 1 = 1 F
3. F 3 23  2 E F
CD

E F 32

3

11 111177770000AB ,,,,,CCCCBC====,A11111111(222233331111BC) AAAAACBBBB ,,,, BBBB
((((((11113333 777777))))))++++++ ((((((22224444111111000000))))))
1BACBธ=====85131(42(513571+ธธธธ+96241553526))2441++8570A((124266=9681111111BBB71AAAA+2+31CCC88BBBB11313))7806===================42((138855551111333311112((((44442222((75555111133335555177771111+,++++66+++++996666B42242424))1111255555543333+5555+222206666))))=))0222244444411++++++88885555((7777200485AAAA((((((111122224444222266666666====99996666888811191197777111196++++222++++3)1111)88888811111113331311113333))))))77778888100006666((713042424224((((((1111333322227777771111,))++++C,,,,666666++++++BBBB))))))2222=4444((++++++00002====40000((((((112222444423888855551100))99999999996666))))))

BCCC 1 =====ธ96A96A96A85131(((4(2(513BBB571111+C7CBC77+9600042155C))3)526))==2=34411111111+1===1+8=5==7====2320311231(A(13ต1(3131(2485323123226่86อ(8(8=9(68((82(1118121821755815)58558+62+83124686245824558+82+96A1+13131+)1)1+1+1718(111106181B3121073210321077)1))242))1822)1C(827)18(22+01+36+++86+86+8+2++)++7+((1(((19=(9(19=1629+91,11966=63+1==33B21311)0)2001341+(+110133=2110218(2(222821(25(258)2)6)468)4258)5+)+2+++)++1+11(11((19(9832101(71(7960)7)7)2))18002+6++8++111((1(1B((B7319113190362333)3C3C1)))0))177==132,))228C)++)6885++=)((249AA(1(9A96A7((1110BB((231111BBCC0010111CC7)0)0)3)3====)==))==)11==============111331==11====66113323238822((((1991336225561113331388((8772811(2(((((955938825234411153135A58(558887(0107881((588582841662884144555585580B808++++82286228+6+8242455+11+++++21122+112111111,+1111+66+188113318811122113300111B771))615568838223130)8)))1100881307))C227)++5)2252))661028++++88)44)108)++2)2++++22,6++846++48++((++++((((((1199A99((662244(((((1933(199(119626244B30031111C001133331221)12213223322322))222))6622662)++))6++)666))))++++)(()))11((((77A1(001(77C0011111331133133))33))))))

===AB858585

BC

3 (ต่อ)

1 2 5 6 7 11
3 4 8 9 10 , C = 12
1 A = , B = 13 AB , BC , A(BC) AC

ธ 1 25 6 7 11 C
3 48
AB = ((=249A13C881))0=24((13131112316246))++1123((24 C2

= 1 42==1112311((5135++1553A6))2+C+  9) (1 7) + (2 10) A
3  9) (3  7) + (4 10) C

6 +18 7 + 20  A
18 + 36 21+ 40
A 23
21 24 27
= 47 54 61

11
BC = 5 6 7 12

2222ธธ====ธ============ธ====AAAAA00002222ธA((((0000000(((66662222AAAA2222000022+++++++2A=11113333ธ===266662222=111122A0000000A33332A===1111AA))))02)))2=20000(222200(2===2++++62+++===A20===22220000======(60202++(=A((((2(((02013+++++++11113333(6002(11113333(6002++(460026(21A22(60020200(22002002032099992==3333=01++1+A+++AA)262)=++3A010230222212221+1+22626223632=132000)113030)02))1612130))))(316002(01301==)(+2))=(0+20)))++=1203)A)02+13++++A+++20A0220+(A2+0((((02AA((029+3+2011A3(0A2(2222A000026(++(1(33A(A0+A+1301(++312(1A3A2A++193313A=93=)13))=)93139313=2911113=+213111+2=20A20222))))2)))2)20202(()(++++A))(++++)13)02A)022002(((((((9313211110=3333132(0((13(((2202211301132013))))110233333331+1+))111))))1)))11+)+)))()(()()++132+(0++1(3+++13(((((13(((131331331133))))A))33+3+33323A))AA)))A))((213222 A2 AAAA 33 ======== A3 = AA 2 3
3 ==== AAAA222244440000222200006666AAAA2222++++++++22221111===11113333666688881131331311911999240206266662222+66660000+1+++++++13683111111113333333391111111133336260 11
3) ==== +11
3) 3 + 33




5 =(AAAA==−B((0BAA1)tAAA−−BBB00=ABBB11))12ttt=(===−A,01122A1B−B0−−−B,,0001)=11BB1t 12===1202121212−−02,00202021B202−02021+=−−−2−1211101−−22−22021112220−2−3−−−1−−133331−−111−−−13233111122 − 1 ((A((AAABBBB)))t)tt,t(,,B,ABBBBtttAtAA)Atttt,tB t At
3 
ธ ธ ธธ A

ธ

− 1

3 

1+ 3
======−−0022−−−−−+−000000+2222220++=+−−=0+++00043−−−−0−0−0−00−3443342064+1−−−−000−0641641641−−−4−−−2−−4344422200164−01+01+01++4++3+6+3366360 6
= +
=

− 2 0 

(AB)t = − 3 − 4

 4 6 

 4 6 

− 2−−−2022−−0−02202  000  Bt At

(5AB(ต()(่อtAA()=BAB)B)t−t()(=At=A3AB=BB−)−)t−t3−t3==4=3−−−−−434334 −−−444 A = − 1 1 At = − 1 0
 0 2  1 2
 4  4464 646446  666 
ธ 2
B t ABtBtBtAAt tAt BtBBtttAAAttt 2 1 − 1 
A = AA−0A=1==−−0A12A0A1−10===11212−−−01020111 121212At =AAtAB−t1=t1===A−AA−102t1t1−t11===1−0202−−−121021111 3020202 Bt =  1

− 1

B = BB02B===−0210B2BB202===−−1−1−230202210212 −−111−11−−−111 tB=tBBAtB−tt12=t1===B−BB−−12−1212t1t03−t11122===1−−−−03−−1203031−221212201312−−−−10303031222 0
−−3−32232  333B 2

 2 2202 020220  000  − 2 + 0 0 + 0
B t ABtB=tBtAAt tA−t1B=Bt1=Bt=ttA−AA−1−1t1t−3t11=2==1−−−3−−−131−1211213112−−−1−02−1313−31212210202−−−11021111 020202= 0 − 4
−1− 2 0 + 6
1+ 3

− 2+−−0−22+2+0+0−−0−+022200+0+++00+00+00000+++000 − 2 0
 −==1=−−−21−1==−=1−0−2−2−−211140−−0−−022−2−444000−−−444 − − 4
=  = 3 6 

 1 +311+1++30331+11++6+00333+0++666000+++666  4

− 2−−−2022−−0−02202  000 

6 AX−2AAAt12AXXX=Xt===A=2Xt2=AA1=13133132222t+2A13A−2AA−−AX−+2222t1t122t1t2tt12AX−=ttt2tt t12t2t2t22tt2==t=A=22t22tt22AA=AAAttt2ttt
(( ) ) (((( (())))) ) (( ) )( (( (()(())) )) )(( )) ( (() ))ธธ
1 A AAAA A2 t t22A2At Att2Attt ttttAttt t t t
3
A= At At 2 tt
2
ธธธXธธ A+

X = 22−−AAAA12−521t++1−XXXXXXXXXXXXXXX7264A23============XXXX====22222−222−−−−==−==−−AA−−1A21A2A−121−52215t−21t22t51t213t5t211tt113−−−1−A1−12−−7256421A72t64A723264−A7213264A232−−2321−−1−72−64A−−−32−1313131313−1133−13−2−−132−2−1132−2−1122−112111−2−211
X =

X =

X =

7

(((AAAA(AAAA(=+=+=+(AA=+A(AAB(AB((AAB1=+3A1(3AAABA13)=+)+A13+)2(++2AB+)2=+++(A13AB2B+B24A)4213BB+B42(ABB2B)+24+AB=13)2=+)=B,)),422+=2B,)B22+42=,==B2===13===1)3B==2413=,B=B2B)=13B,12=13=4B=3414)=52134340==521=,0=45042522345=0024=B1345052240=+=+1B=50+24+34+=13+4512+0=12341+12B+50522414+525212210=342+5234150042+122103344+5+2504502552+34500=50+50042121+52152501012252103+41215122+50+21052212345105052011211+52515042124340101502041001012525140505102520+52+1+10++5212+410152+01+10142521021010+210+5250425+050+1015522+10+2(5=(0=(1A=5A(2=A0A5(+=52+52+(A52=+52ABB(=B+52AB52)+525)252)22B52)2+52B252)B252)252)2 A2 + 2 AB + B 2
A2 + 2 AB + B 2
ธ ธธ ธ AAAA222A2+++2+222+2AAAA2BBBBA++B++BBB+B222B2 2

ธธธธ

7 (ต่อ) A = 1 2 B = 1 0 (A + B)2 A2 +
3 4 , 2 1

ธ A2 + 2AB ธ+ B 2 (A + B)2

B======+A+++A1133171132B35222712BAAA5++++2AAA++22222412BBB64211222======6===122A02A+++AAA02AAA13B13111B11333332227771112226BBB5552+6+++2++++++++++22242++B424224111+B2226662221AAA1112222221826AAA82AAA0002226=222=111333=BBBBBB==========666222+++1+++7AAA17+++742422451117111333+++333BBB5777111222BBB555111+++888222++++++6662221=12===2224441111A122202666011122220222022AAA000222111777111333777=BBB==++555++666A222+++2113231+++422244711112+++111BBB522201+00011151++0002221A58882226661A4212=B6===122B+4+++A04=2=42===1243B222=111777111777+55561+31+2+000111131351+5551315+11142AAA+1++B1+11111411BBB122248(14A0008244442480006222A42===42===224244====24=+00+10+00+++101211133300111+++1B111233315557B++71115222++++++5+111=)111444241000044421118884214442220555==111AAA4224242=131200002BBB01110002000000144411134222===452===4424220===5024+++++++++++111244242333111+++11100021113331555+5221111342++++++01115050121114445444888121A44242252442422B10008441110000=000000=421011=2225++B+++++++B1311+13225424422111000012++5=14==1=4=48=42421=1212101412+BBB10+1400++125222202++1002===1010===10===241052012011122201112220111444++++++++10100022211100010111000BBB222000111222000=========++++++111222111000111000111222111444

7 (ต========่อ)111133333777111715555++++++++41111666611112222222200002222ธ3 66662222++++4++++111188886666 A = 1 2 ==== =111133331111355551111++++++++124444888844424222410000000002++++++++ 244422421 ====
3 4 ,
B==== (A + B)2

(A + B)2
================1111111111111111777744447777777755555555555588881111++++++++ 11111111222222221111111122223333(A00000000222222220000444422221111A+++++++++++++++++11114444B2222B11112222=0000)111122221111555522222111100002222===13++++88884444++++44242242144443488884520++++5042++++++++++++++00001111111144441+52111144442340550111100001211110000521401052++1205
ธ AAAA2222 ++++ 2222 AAAABBBB ++++ BBBB 2 = 2 2
222 5 5

ธ 8ธธธธ ธAAAAAtAธABttAtttAABtAABBBBtBt2tttttABA+tttBtB2BBtCA22+2AAAAAt+++ttC+2+ttCCtBtCtBtttt22+tABAC222+++B2+BBBBt2AAtCCCCtCCtt2AAtCt2t2+t2ABAt2AAAABtt2CAtt+BCCCCtt=ttttttCtBt2AA=BtB=B==t=2=+C=Bt=tt=====tttC=tB102=2t2=AA1022=102=101202=t12022C1=0=22=tt110022102122B110222=210=2124112222102110222142==t110022544112424102514424122544544544102−1022=142−14210212−−14215244−124142142188721424878121224−878487810248871A24142A12A544AA4887A10++−==10+10−+A05−=010−=+1=2412−=050511500+478814241244−=1211214242442504241A2121244212121211212BB44210+B12B12−=B−−B50t32t−1212−4242−t3211B2t32241−2t2342812824=t23=24412842−8=1212=8t32=12842=048B04=041004−10104−101−−10041−0==t13201120==−24,01100==04,0−0==21,00==28,010=2−,00==12,20120==212,0102424121210224241022441204102424B12102424B10−1024241B12B1022440==B23,0=B523442−32=32544=−544−=3224544=−544=−32=241242=8=2453244−102244==2488544−1238=24B81233=427881238544123878878858874454412354478832132=−5441878544−−5−88744110021−544=1240−100210221002080−−1100022−21−02132−781002,−21002078−,8780874,2875,442−784−,4478C,78−4,1CC41C00240C12−=CC18611=7876,==186786=77486117==86861227C712211222212211221=22867−−12−−124−1122224−1−2441212444−−−5012−05−150−4105−051−10515011−051

ABAB+ธธAA+ACBCธAt AtธC=B9A=tAB=+12BAAธAA12+AธBB+CB−AC−2AAA3Bt+3+A−1tCCB3=AAAC1+AAA=tBtCB1ACtC−B3==A1222AtB+−1=t1C2t1=B+AAA1+=A−12=t−AACB224+A3C12103==A12−CB1t3C−33t1A2C−12=t+331−21tC−1−=−=t33222330123−=12=1t−21315A1−12=−132224342−1−112110−=1−3222A31−333−1−1−1,2243+−3A101=13231−−1133324==310210=−13−+222B12533−1−1−−143+33−−2−022A112=1−8−2−25115423334−02A101=4−3,2−5133−423−110=413+==11121−,33−3−3B233+021112−,2−==−−−=533382−53B0241333−42==241−5−−−B−=8−84253=0123,3341−−−−11=−3832532−1,334==1,1−−1137B−33−3C13172==24−11−−1−=83B2538233=33=40213−1−−24=−2833−25518,334−42=02−11303−−37382−1C7=02113,1−1−1−33327=321,C7−42−11−12728533C72=0=2−303421−12=5,81−=02−130327251C17−−,031−12−122=71C1−7−1212512−−=0312112251−−

ม ั ิ องเมทรกิ ซท์ เก ว องกั ก รคณู ละเมทริกซ์ ลั เ ล

A, B, C, D ,E c

1. A(BC ) = (AB)C
2. 0rm A = 0rn , A0np = 0mp

3. I m A = A , AI n = A
4. (cA)B = A(cB) = c(AB)

5. A(B + D) = AB + AD
6. (A + E)B = AB + EB

7. (A+ B)2 = (A+ B)(A+ B)

= A2 + BA + AB + B 2

8. (A + B)(A − B) = A2 + BA− AB − B2

9. ( )At t = A

10. (A  B)t = At  Bt

11. (AB)t = Bt At

12. (cA)t = cAt

1223x0ธ((+3xx2222233xx6(223x((((3++++y33((−xxxxx666+3((xx633(y+++2++yyy−−−3++2Ay)A−)z2zyyy−222)yyB+By222yAA))2+2AA2A))3xzzzA22zzz422zA)−−−A)))(−z2(BB(++=zBB=−B+B−2+()3++xxB++B24446−(4+C((AC2==(34==−−=++22(=−((y(2−(=222=−−−x22(−(=CCAACC22C2AC−y2z−((2y+CA(C2322xxx2)2yyA)(((==xA)z2(=z22xyzz−−−x)y(=+++z−B+333B+)))z−3yyyy+)++)yy)+32243yyyxx)2yyyyy(yx=ay=−2))=y(xyy))+++))+++y+2333)+−))+yyyz(3CAC)+aaa2y)+==3=xa(yyy2y=yyy1+++02ayxy=+zzz(((=y+yz(2y+3zz−xxx222+z(x1110032)yy1y0x++2yyyy2222331a10zzzy+xyy23z)++y)2yyy3z)+,yy=)+ya11aa13yyB1ay))+++ayy+=1a)+y,,,yy===zy+)+,y=10=zBB(yyy,2y=+++Bxy2zz+10By111000y===z++222y10−2=z3z1a210=2yyy2y2y−−−y1a111aaay−222)+=1a−2222,y=1a22B===yyy2+10=z1=0=yyyy211001yy10y,yy1a10−C1a2,,1a1a2CC=,a1,Ca1=C===10=yy10=11001010a110,a1aa11C1a1a1=10 a
1

ธ ธ ธธธธ

(x 2((+(xxxx(xx222+++y+2+((+)3(xxxyyyyx6yy+++y))3333++−)yyy3(y+666)6yyyyyx6yyyy−y2+++2+−−−y−2+++y))++−yz+)4z(−−−222−22222xyyy−2==y2y==yyzzzy444yzzzz4zy+z)340za11==============y=6=y====y000aaa111111+0a11−+)0a11…………−22……y…………………………yz……4…z…………………………………==…………==………………………………………………0a11……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….....……….…….…….………………((((....…..…..……3412.............).).).)...…((.…((…((…(((....333441122((((34…1…2…(…())())())))3412))))…………)))) …………........ (1)

(2)

(3)
(4)

zzzz====3333 zzz===z3333(((=444(4))3) (4)

3) 10 (ต่อ) 2222xxxx222+++2+yyy111333y222xxx1===32xxxx=−−−222+++x−2111666+333+++xxx111163+111666xyyy333122216−−−666222y32−62+++2−−−111+−−−xxx−−−333666−1333−444x−xyyy363===3333334444444yyyyyyy=332+44y======y111=================y6661====44400013=21116666x11111111100011140=16(((111x01111−(2+11))63+x1)16y32−62+−1−x−3634y=3344yy==1=====6401611101(1)

(3)
2222zzzyyyzzyyxx======xx====2++333++111666333z33y1166331116yyyx=1166=yy++++3++163−−666zzz−−333zzz1666yyyzzy33zz44yyyy===yyy===44yy==yy===+===========333−====6==−−−−−−z333111(((666z11111y−−−−11144411((6611(((114y11=y44666222111=(()))=662211))=)))==32))−−1(611zy14(621)x==)+316316y2+−6z3zy4y=y=====3−−1(61114(621) )

x == 14 10 (ต่อ1111)66646444y++1++5121521=5265266565516516(1616((16(221216211161156)1615)11625)511)62151621616251625+165216+2+2+x+x++++++++++21+++x2+2+2==612911249122142924=2262422426216112622166216+166616=141465211=61=616=61=6=1=66=15=6=416a=(a=a=aa2a==aa1aa=16===a5=)1=a162=aa516a2a+aaaaa++a+yy+21129y=22=426216=13166=161=6=16a=aa==a11a6a644164++152152665+51165216(6(25211616116(5)15162)121625116165216+52)+1162++516++a+2+a++21+2+1+129a+1=22924+2=22214621626212916=2244316=2623416=1=6=161=6=4316=a=1=6a=aa=aa===a=a=aaa=a=aaaaaa
x == 14
xx 116644 xyyyy=====16161616146
66

1 2 0 1 0 3 − 2
2 1 4, 2 1, C 2
1) A = B = 3 2 = 0  AB, BA AB + C2


1 3 −1 3 2 −1 2 
2 − 1  1 3 0  
2) A = , B = ,C =  0 1  A2 − 2BC

 3 − 2

3) A = 1 1 2 X
− 1 1 3

3.1) AAt = 2I2 + X

3.2) 2At A = X − I3

4) A = 1 − 1 AAt (AA)t
2 4 

1 2 −1 −1 1 4
A = 0 1   3
5) 2  ,B =  2 1 (AB)t At Bt

3 −1 1  −1 1 2