3ม5-6-64

เมทริกซ์สลบั เปลย่ี น

เมทริกซ์สลบั เปลย่ี น

 A = aij mn  B = bij nm

bij = a ji i 1,2,3,...,m j 1,2,3,...,n

B (transpose of a matrix) A

B At

่1

A = 0 1 At = 0 2
2 3 1 3

− 1 0 1 − 1 − 3
− 3 2 3  
D = Dt =  0 2 

 1 3 

( )A1 2 ่ 23 1. AAt=At =14 12 52 6363 ( )( )AAt tt t
4 5 45
= 6 1.

1 4 11 44 ( )Att = 1 2 3 ( )Atttt = 11 22 33
At A=t =2255 4 5 6 55 66
At = 2 5 ธธ ธ 4
3366
3 6

1 4 −1 5 11 4 4 −−11 55 ( )(AAt tt )t

( )2. A = 2 0 2.2.8 A9A==22 0A0t t 88 99
3 − 2 7 6 33 −−2 2 77 66

 1 2 3   1 1 2 2 33  1 4 −1 5 11 44 −−1
 ธ0ธธ − 2 AtA=t =−4−141 −−22 At t = 2 0 8 AAt9tt t==22 00 88
 4 00 77 
8 7 88 3 − 2 7 6 33 −−22 77
( ) (( ))ธAt=
− 1
   5  
 5 9 6   5 99 6 6 

่ 2 ( )At t = A

เมทริกซ์ศูนย์

เมทริกซ์ศูนย์

่ mn ่ mn 0

0 0mn 0 0 mn 0

่3

่4 ่4

0 00 00 0 0 23 0 23 0
0 00 00 0 031 031 0
013 013 0
0 0 0
0 0
0 0

0 00 00 00

การคูณเมทริกซ์
ด้วยสเกลาร์

คณู ด้ว

  mA = aij mn r

 rA = raij mn

่4

44 4A = − 12AA=A=A−11=−=−−−1212−−23−−1212−1−111−−1111232323223A 22AA22AA
−

ธธ ธ ธ2A = 22A−A−22=12A=A ==2−211−−22−−1212−−23−−1212−1−111−−11112323=2332−− 2 ==−2==−−2−−4224−−−−644242−−2222−−222264646464
4

B = 1 BB−−=BB=12=12=12431212−−−−1212−−−−12124343−34342B −−22−−BB22BB
2

− 2Bธ = −−−22−−2BB22=12BB= ==−−−−2122−−122212341212−−−−1212−−−−121234434343=−− 2 ==42=−−=−−−−4224−−−−8642242442 44−−66−−66
4 22−−88−−88

C = 5 CC=CC=2=5=555 22221C 11CC11 CC


B =−−2412 −2 −32B− =2B−=2−−21B2−12− 2 − 32 3 = −−=42−− 42 −4 6− 6
= 644 1 −41 4 24 −2 8− 8
−21 6
B= − 1 3) C ==C53=−− 42534224 2− 8 1 C 1 C
22 −2  4 4
4
−่ 42( 4 −

−1

− 2B 
( () )=Cธ=−−  1144=((535))1144 ((5314))142(414214) 2   5  52  2
( )1 C = ธ2ธ53 4 2 1 C 14=14CC=14 51 5 2  2  =    =  4=   4
4 34   4     3  434  1
41 C 42 − 6 34    (4)    41 
4 4  8     4 
−  
 1 12 
1 5 2 =  (5) 4  =  4 4 
   4 (3)   3 
4 3  1 1 (4)  
4 1 
4 4  4

( )=1(5) 1 2  5 2
 (3)   
 4 4 (4)  =  4 4 
 1 1   3
1 
4 4   4 

การบวกลบเมทริกซ์

ว คูณ ด้ว

   A = aij
B = bij mn

 A + B = aij + bij mn

 A − B = aij − bij mn

่ 1่ 5่ ่่111 111.A.. A+AAAA=ABAA+++=B==12BB12,1122B−− 12+−−−−−−12A1122243.,2B2243,.4433−..−,2−,,−−AA2B22AB=AABB===10A10110023− 232323−1−4−−111444
B
1.

ธ ธธธธ A +AAAB+++=BBB=1==2++12112201++++++010011−− 2−−−+2222+++222 +3444+(+3−++331+(++((4−−−1)11444)))
1−−−+1113+++3334

= =1==313113302 02002204040044

− 2) 1+3  − B2−−−A+222=AAAA−====2−−−1021222++1211221−12−−−213112+22+43((−−43433412)) 1+3 
− 1) − 4 + 4 − 4 + 4

= 1 ====−−021243−−−−−−420424420242 24242404−−86−−−−−−868866 = 1 0 4
=3 3 2 0

่ 5 (ตอ่่ 1) A = 1 −2 3 B = 0 2 1
2 −1 4, 1 3 − 4

1. A + B 2. − 2A

ธ AABABB−−−−+−−2BBAABAA=============11111−1221103−−111002−−−−−2−−−−−−11−+1111−−−−+0−−0111221220211014444444444−−−−−23223304−182218−221−−−2−−−−−−−−−−−−21−((82((((8822−3−+32−−+2−−4312113222))))))44 −3−34−(−(+−3−11+(44−))14) BB −− AA == 1100−−−−2121 222 −−− ((−−− 222))
ธ == −−−−1111 333 −−− ((−−−111))
−−−111444−−−−−−333444 −−−−−−828822
44
44

−−−222 444−3−−33−(−−((−−−111444)))
−−−333



= − 2 4 − 6
− 4 2 − 8

1 1 
่ 6่ 6 102AAAA++++−1B2BAA0A10A+A++==A=+0A0+0+B1212A1A+2=BBAA0+A+=A−12−−+01++01=11+111020=B=BAAA−A121=03A031+031+2+−=101210−+BB+1+0312A−1=100031A−102+102B+1A12−031+B1+00+10300−−B1+2120A−03111102+00031+00102+AB00−่+013200001−12=102001A+00=1200−00=B12A121A00−112001−=11−00A031=112ABB031203,−=1BA1−1B=121B030203−11B−2103

ธ ธ

= 1 1 0 +
2 −1 3

A+ B

A +0

A + 0 = 1 1 0 + 0 0 000=A12+ =0−A1121A+=+003−A101A+=+03=A000 A = A = 1 1 0
2 −1 3 0 A 0+ 0 0 2 −1 3
0 0

A+0 = A A+0 = A

1B+ 3 ่A7 ่ 2 8 1 1 6 4 
2 A = − 5 3 2 , B = 3 
− 2B, 2 16 22AA22+A+B+B, ,B3,43AA3−A−622−BB2,−,B912,12BB12+B+23+23A23AA2 1 5 
−1
81 ่ 7่ 7่ 7 ่ − 1AA=A==−42−2455−245863863 863−1−21299−12,9, ,BB
32 ่่
ธ 4
ธ= −1202AA2+6+AB+B B4  6
 ธธ  2 8 1 1 4 
1222AA542−AA1==8=2=2−4−2524−245=5−24635−86358637−8632−91−22192792−,129B96===23=−−84184−10−10184101161126−6562111626−−4212418−84218
− 26B, −19B + 3 A  ่ 8
4 162 2 2 1
 3 6
10 6 4  + 1

88 112 − 18  24 −116 −12  B ===−−8418410−108401011611266261111626−−42124−18−8124198+++132123132−16−1611−161−54−5141−541 ===−1−5105707−1507127121271212712−−9619619−9
 2 = −10 622AA2+4+AB+B   5 22 6 
3 2 
68  =1−8613563A1A32−29A−4422−B−B21B863
3 1−6192

06 6− 9 4  + 3 121 13−835A11A830−A 2=7=33=−=342−4255−−142508673863 8631−71−211299−129−919===−1−61126125−156125129128498412984−−632632−776327
6 
1 18 =262
−1 124 − −112
5  2

− 1 4 − 2 − 2 11 6 6 4 4  22 1212 8 8 

32,=11−−−−654621223124219595−A2841−−08342863128ธB112=−6,=62=12+12912624−123−1−B6814123241่ 50−137+−8A2162A23133(32−1211ตA==−9333+226AA842633222A2อ่ AAABA−−21−AAB−AA54−8B33)223−201332−2B13322−2−+2−−−2A+AB336AB22A2814A1AABAB242B632่1AAB5BA20B2B−−++−B8=−7=B+===B16−+212BB3232B2211=2==B=BB96====B462==B−−1−321333−2244242===435442−352==5=511=−1−−−=61123=16−12242842−41−48441231−132−5−1−15426554263411164051−6−A863−−01063856636117618112−−284841−616−115225=166841168421−01−08635186392511166166−116−8−49011−6863018−6813−5624624121−62754621921−911229121991142111992−8−=4−1−6−28−−694115−546212462−89−1112454−6−−8−−14546291612991639−54962−1229142919−6311927−249611−85863−−0−−7−79863−−62342632142263−1=4221=−247=1+7638728=724−81=−+112278762426249=−132−12==−+1==−=+=−6=24−6+64281−11232=1,=−5+06624−816421−−−62241123−642−662424−13212662442−12−6123251261112−81112−2696164281−1662428−18121B321221252621−259916−2511−25−86421−−1211−−−211261−−2112−−21=21262−−2212126−12592112622254−8281224218−2211102−121012292121261−−21−18−21−9422121235489124822880−42−416310−111222−−524−1−8−1−9−81−−5410−848101874828540281008−−2206321210−−2−2−1−−−2116−54−8638187201610311=22286317−288787=632=−−547=84=12=−=1=51−=0=7=−84=−=−2−184−2814−51218420−7=21−517841212002517−−7511012075170841272−171212−21025171−7−2012270−71722701212270127547961212205127191−22−7−−−12−2−−0−−−−72−−−1225496−1−254965−1−5496251−95549619259−549619−5−−925496159
244 36 − 92 12 812 18  −42712 − 5 
 64 −16202 = 180−821 
91 66  −4 2 12 7 − 4 
2183 5 − 2= 2 20
−127 − 25

2 −1 −1 4 − 2 − 2

2 8 1 1 6 4 
ธ ่ 7 (ตอ่ )322,=−1−234234255−A841−−16086318632B11−−−6,5462121219912−B241+81232112212==111222=B+BBABBBB+642++−123−+++1+236323281412232323A่5320−AA1−AAA2162112232A211111222322321211111111232222232223BBA96BBBB8AA46211223BBBBBBAAA+−−1++548+++B0A231=22323=−==−322323==A===12===42AA6313221122AAAA3232=111222873323322===123====−=11233242−−111232323113224242−5−−1211112323222424542532−11111123223222315655−−−116614−−−=−863111666111325−8−86633111−132−8−−886663331123321242−1321111231233222554−−−−111122255441−−−1−2−56−3051551444112912−−−67−111111992221−−522999863−−−55222212−1555112222223221111222227==−912+==5=4=−++======112+++,−113229−−−111122332263−−1−−−−111111223232322636635322−−−1115636326662233311961111555−666333B2222211155555−−9222123555−−−112312321===11111232329932221112+9911199222221119921122299999922299992211122222−99999922223−222−11−322−522−−−−−5−225122263−13231−−1−555222222222−−32133223321112222257−−−233222323233233−2227323753232212−222777723233332322222177123222777121992129−25===1−===−522−1277−−2332777722−−−627777776622227323666
 −1−−722111−−−177222211111417772222221114411144472 15



111511155555
555555

111711177777
222222


22B++BBAAt2BBt2+2Att2AAB+==A2+B+tB2A+tธB6B2t2B14=A==−4่+8924ธ12่ ธ6B่2ธ81ธ415203882−4A2่94212ธ1A+Aธ40A=+2252B+03A+AA22Bt+22AB28222AA=2B++tAAAt=22BAAA62t++BB++AA22+1Bt=1422B++8+8BB22tB1A4A2tBB22A+2AABtA22BB−BAAtAt2AA824A+t22t9=4B+2AA1=t=12tA+2A94+A++4+BAt0==B22A++B2A==1+22BB14+B+tBB=8=56B22+AA03tt82+BBAA+2t52tB1426B2tB2=26B+22B,t2B14A+26B294A1=4,−A1=AB144A4==่==89+240B−1+==+−t4B+่894+24−252t่1B8924B41B21่6B28924B+=1146B522034814,=8=1415203=185203881024−5280384−11่80428924B141่8972244101=1=−4046311A0=−520341=81163052+8038=48A2183024A=+A+8282=6+2=1821=1441628−04628=63011462=18114A−A8811484A8+94A8−812A1+−=4824A944−=01941411629441A04152621030144810148528035280B3,A−5203A4−B,94B4,11=94B1,410B4=0=BB152=03B=452034B11=4B,=441B,4=4024411=024=70224B1111042B72−1117263=41724−1=14−634463−31163110424134104211341137272−63−6311141343

1 4 2 2 8 4 

At +ธ B t 1462่ ((9่24((AA+AA++(+++A10(BBAA((BBAAA+13((AA))tt((AA))tttt+B((B23ttA+A+A++(+A+A++02A)tAAAAAABtA++BB(AAA(AAAtBB++BB=+AtBtBtAt)At+++tt(ttttt)B)B++ttAAA+=+=)B)+ABtt+B+AA=++==BtBtB))13AAttBAABA))+ttBB+Bttt)BB=ttBB1111tt+===ttt==111112A+++tt==tB==))B++tt0=2+=ttB==tA22BBB11==)112224=422424BB=+t1122Bt2424t424214tt1124=1111=tt=+t1111B162424241144,6622=1144Bt6622122411426่่22424่่2424t++1114B62++++่421462++=1100+่4211001133++11331010+BB2233BB223313100022tt0022tt3213B23==AA==AA02Bt23==1133===02At1133=====A1122=13112200==22=00==222222=132222121144=24420=211441422422266440=26644,,1422,,2466226622146442,BBBB6462,====B62B1100=1100=1032322323102323

+ B)t

A + B = 1
2

A+ B)t = 2
4

+ Bt

At = 1 2
1 4

At + Bt = 1
1

่ 10 ่ 5 ่่่ 555 ่5 =2XA−XXXX+2322====X22222(XXXXAAAAAA−−−−AX=22+X=++++32233223+2A−22222222−2X2222((((XAAAA+32AAAAX−−−−−XAAAAX3X22XXXX0422222++++)2(====A++++222222222A==−=A่==−−−−X2222XXXX2+XXXX=+22XXXX62−XX−−−−XXXXXXXX3333112XXXX2−000044442X042222))))X322222+====+XAA========−X่่่่====X=3====X042)A1−21266662−2−22−−XXXX2XXXX622=1111111122222222==่==00004444−(22323A3223A่ ++++3++++AAAAAAAAA62−XX====1122A304AAAA1111−−22−−221111+222232666622222222++AA=−−−−((((2222AAAAAAX่AA่่่A33331−2AA1AA−2622AAAA33330−)(2++++6AA่ 3A=A3XXXX−−−−+X0000−−))))6666X−====+930)6XXXX12=−−−−−−−−

ธ ธ ธธธ ธ X A++++X 11112222+ AAAA12 A

่ ่่่ ่ 99993333−− 9
3

ตว

A, B, C, 0 ่ mn ่

1. A + B mn ่c
่ c, d
2. A + B = B + A ่ c, d
3. A + (B + C) = (B + A) + C

4. A + 0 = 0 + A = A
5. A + (− A) = (− A) + A = 0

6. c(A + B) = cA + cB

7. (c + d )A = cA + dA
8. (cd )A = c(dA)

9. 1A = A
10. 0A = 0

1. A = 1 1 0 B = − 1 1 2 A + 2B
2 −1 3, − 2 −1 3

−1 0 1 1 
 1, B = 2 −1
2. A =  2 A − 2B

 3 3 0 2 

3. A = 1 − 1 2At − A
3 2 

4. A = 1 − 2 B = 2 − 2 C = 1 0 (A + B)+ C, A − (B + C)
3 − 1, − 1 − 2, 0 1

1 2 − 2 3 0 − 2 −1 4  (− 2)Bt , (A + B)t − C t
− 1 1 4 1  
5. A = 0 , B = 0 , C =  2 3 
 
 0 − 3

6. = 3 1 − 3 = 1 3 − 5 X 2X − A = 3B
A 4 −2 , B 0 2 
2  4 