9ม18-19-64

การใชเ้ มทรกิ ซ์
แก้ระบบสมการเชิงเสน้

กกาารแรแกก้ระร้ ะบบบบสสมมกกาารเรชเชิงเงิ สเส้นน้ ใชใชเ้ มเ้ มททรกิริกซซ์สส์าามมาารรททาา ้ 3้ 3 ิ ิ
11. .
22. .
33. .

การแกร้ ะบบสมการ ใช้ ิ การ
าเนินการตามแ

(row operation)

การแกร้ ะบบสมการเชงิ เส้น ใช้การ าเนนิ การตามแ ้

1. 1. AX = B AX = B
2. 2.
3. 3. AB AB

3.1 3.1 22

3.2 Rij Rij i ji j

3.3 3.2 cRi i i c c 

cRi

3.3

3. 3. 333333333.........33333312132132......33113322..333...132333...333123...123 cccR333333RRRRRcc......iRRi113322ijijjRRiiiijjiicRRicjRi RcRi2j22iRi22ji 2 22
3. 22
ij
RRiijj
ii jj i j
ccRRii
ii jj i j

i ii j jj c c0
i
i i ci c cc00 c c 
i i ci c c c0 0 c c 

i ii c c c  0c c 0 0

RRiRRi ++ii ++ccRRccRRjjRjj i + cRj + cR j ic j j
RRii + cR j i i cc i j
i i cc i cjj
Ri + RcRi R+j i c+RcjR j i i ci c c cj j

j jj

4. 4. 3.13−.13−.33.3 AABB

I PI P X =X P= P

3322xxxxxx++32x−−++xyyx+yyyy−+++3322xxy−−−−xxxxzzyy+++xxzz−−++−==−yyz====yy55xyyz++=9944−−−−==zz5xx9zz4======559944

11 123−−111111113322−111−−−−11−−11111111−−11111xxzyzy−−−−11x1111zy===995544xxyzyz 954== 55
3232 9944

113322 132−−111111113322−111−−−−11−−11111111−−11111 995544−−−−111111954 5544
99

ท บท AABBAABB  II P PIPI P P
PP
ท ท บท บท AX = B

AX A=XB= B P

เม เมทริกซ์แตงเติม ม ร้ ะบบสมการแ ะ เ งต างต น
เม เมทรกิ ซ์แตงเติม 1ม 0้ระบบ0ส2มการแ ะ เ งxต1 = 2างต น
(2,5,1)
1 01. 00 21 0 5 x2 =x5= 2
50 1 1 x3 =y1= 5 (2,5,1)
1. 0 1 00
0 0 1 1 z =1 (0,3,1)

1 0 0 0 x1 = 0 (0,3,1)
0000 3
1 0 1003 0 1 x2 = 3x = 0
0 1 x3 = 1y = 3
2. 0 12. 1  0 = 0z = 1
0 1
0 0000 0000 0 0
0 0 0 0=0 0=0
0
0 0=0

3. 3(130000.(.1000031000010000(.3(310000.(100001.00001000010000x100001000010000,10000yx10000,10000,x11003z10000,yy,1000013xz,1000011003,zxyx,,,11003yzy,11003,z11003z 0 =010==)101)=0)01==1)1)) x =====0x0yz01100x30yz==========110030x0yz11003==0x===y0z0x0yz=11=00=3=======1100311003
( y
z
0
0

( (( 4. (4(14000. .10004z10001000.=44z1000.r1000.1z0021000==1000r10001002z1000r1002=z1000z10000043=r1002=r100234r10020043 003400340043 x =x3x=−=3x23−r=x−,x2=3y2r=,r−=3,3y2−y4−=r2−=,x24yryrr4+,−00+,,x=yy−z2yxr==z+y=400z+,r=+==00+00z2,−=4==xzrz24yz===4zr−+3=z00x=−00+=,)yx=00ryr=z2+4r==00+z3+,4z00+,)=3z2===00zz2)===zz=rz==400=3=00r=)r4343))
(

11
1 2x +2xy += y4 = 4

2xx+−yyx=−=4y−1= −1

x − y = −1 A BAA BB
   AAABBBA===B1212A=−−−111B111212−I−~~44=~4~~~−~~~1111P121111−1112111100010II11100101−PP−~142~~−−~1−−1−211−−1220041211−12011221A121113−1100130−411552222B115221−−−1−555~−−4−~~~~−1~~2155−−11−12R0−5−R16642611~1~~1131~1133RR11131R01011010−1522−11RR−RR1RR511−22−1010122021−22−2~5−2−~~R~21−RR−12022R6−1RR22−1R11322R3325−2122211R−−1210212010R10−15221RR2−51−111523−5211252−−2R1−56−RR−5−−1R12610213115R2−2R113R−26R3R5212−1R11R1−3R1R225−−2−2−2R215R1R−R−R6R2−R2−2112132RRR22−R−RR111R21R2222−221R−R22R22 11

IIIIPPPPxI =P1x =;I1Py;= y2 = 2

x =1 ; y = 2

 

22 1 0
~ 0 1
2x +22xxy++− yzy−=−z1z ==11 0
1
x − 3xy5xyzz−−−−==3535zz−z−z8=1===3−A−−−8181B33A~~ABB110000 ~ 0 1
y − 0
00 −− 331 100−−−−11337788−00−5130RR33 −8  ~−1378100 0
11 −−~55110000  1
    AAAABBBB 00 −− 17  R3 − R2 0
RR22− 30 

====~~~~~~~~~~~~~111100002222111100002222111100000000111100000000102 111111110000102 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−1021105555155550−2−111155553333−11111111555533335555333333330000153 1−1 153−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−11111111−−1111−−−−−−11111111−11118888111177771188338383133337757788388133338888815333338300003−−−RRRR−RRRRR1RRRR1111111822221383333~~~~23−−−−−−−−2222RRRR100100102100~~~~RRRRRRRR~~~~2222222211111001101000102111000110100100102100 0−03 −−−−−−153181533−−−~~1−1R1813813110000RR11100001R2−−1155R33R~2 2100 −−11337710088 −3 0
0−0 1−11
111111110000110 1−1 1−15 −− 511 RR33 1
111111110000 11100 1
111100000000 − 01

0−03 −−−−−−51553178533−−1−−~~117R8173823110000−RR221100002R−−121100002RR~1 1 100112332 RRRR101122 ++−−003355RRRR3333 1 R11 +03
115 0 1 2~ R02 −15
1−15
;Z 3 0
100−−1005130II−−−−−51−1PP5310373800−−1−−137R8370830−RIRIR3Px32−P−=R1R2 2 ; 0

y=2 = 3

0−03 −1−153715383−1−−3173871810−−R11301 RR3 3
115
001

      3 3AAABABBB3−3x=x~x111==3−~~−~+~=−~100−13xx17x~~x~~11−2−~~1001100100−10011313x210013−1+−1x1−12−1+0011−−001001100121100110071x1000−1−+−2072−3−−−2+−x2−−−−1113−11111−−x−−111x41000x1−1110002−110+0310010x70−217221−1072311x111721−−1+2347x11172+−52+3050=3−0−A5−2−−4−2−−−−x−342−72437=731−−243x7275725=258532x2437B7522−555225x2x2−2−115−32−23x−21−=123212A0+2118522520=18520=+81234B−2+1−117~~~118−−18118940x−10118110131180−−1−04−−~37−7x−77209−x7889981−11−11110010000318941−91−=1−3~~~A144−74−−70094−978=810001198=48R41~99423BR444914100100−1101000R3112−RR081RR11RR31−0−10RR+3R3−R1R2−032−+1R2−13R−−1001003−~3~~−−R2−1−R3+17−+0RR11R1I+7I11+R+1515−1~33+05R12−3−2−RR−20−211001000R−1P1RRP11R−−R1122111−0071R12R11−0521000−72R7155952RR25−21R122−2−10010022−−7122091182−152−7499−−5204−119811−2−−~1−−7710409781551−45~~~~~~~9−R−2−400981−12RR32−452R1100001−111000000001007R1R13952+−−1R312−2−−−2x−17R1R011000010019100001−00131−801−R+R04RR11139−1=I+49833I10R22−2R100−1−00−−R31−−4−P17P17−2175715+55501R525R−2−−23;R2R2RR352RRR2R131132x1+321322−2−1−11−−−22−37−=7+909~RR9R1R−1R1−111−31R4−71271202R9098~~~~1R4−0094+−2−+−14−−;1757R51100001100005R725RR1R1RR2R−Rx32R32−−2−33R333R31R−3R−110000100−1++R4310R1=3+3+311312−1R2−RR110R210100−1−01R10−−R1177~5R55~~~R2522−2−22~21005211001000013200−1100100100−−2−7

44444 3322xx32x32xxxxxx−−xxx−−−−++−+33−+3322yyyy22yyyyyyy−−−−y−−−−+++33+zz332xz3z22zz2z==xx2z=−zz===zz=55−=+5==53=−−=−−2y−−22y−2−2y333−−3+3z2z=z=5=−−23 ~~ 00 77 −11−117777110021111002−11611−7716117712751527515111133~~~~~~~~~~111111000000000000−−−11000011000012−12−1763−−1763−−−7375110000−−−−7375973937939175371717777715211110000000023333RRRR−−11111R2R1R27777R−11110000221111−71R−−R−−66−111711R771177R−31−−313227715511551111321633333R2136+−21R1+−171772712−−1−2R−2R237RR37RR233RR233R33R33−−R−−R1122RR2222−−2211776633−−11−−−−11117733775597973399337171552222RRRR11RR22RR−−7171RRRR−−−−3311 33221166RR++−−111177772211−−−−22RR3377RRRR223333RRRR
00 77
AAAABBBB~~~~==~~~~=~~=~~112332A1231132231111111230000000012311100001231110000 B−−−−−−−11−1123322323= −−−−−−−221−322211331313 −−−−−−11717732−−3233−−−−−−−−−−−−−−−−−−55−55−−−−5−−5519779−33799533523327559739359733232239−32222−322232232−−−2−1221752513111333RRRRR1717RRRRR71RR7111RR111333322RR33233R2+R+−−−−22−−−2−23−3−2233−−−−235R253R5973RR9RR323xRRxR23R222111111=11=1I1IR17RRP1P;;332R −y−2y −−33
−−−−−−−−33222332~ −−−−−−132−22−2211331313 11 11
~~ 00 77
1111 00
−−−−3333 −−−−13333 00
00−−0−0−77117777177717773333~~11112277−−211112−11−7755551100001751151175533111133 11 11
 00
~~ 00
 00
00
 11
00
11
 00
~~ 00 11
 00
00

3R1
2R~~11010

00

II PP
== 22xx ==;; 11zz =;;= −−yy11== 22 ;; zz == −−11

xx11 ++x1xx1+22 +++2x2xx2x33 2++++5xxx5444x==4=11=122717
5    4 4 x1 x+1 x+332xx+112 3++xxx+x1133221xx+++xxA31+2222ABx++2xx++4Bx222x44=2x4++xx12+=33552+41xx−−244x4xx3x==443−11==−77x334x114==3131
     x 4==x~~=x=rr3xs100=1131003==s113r3122−10011rx113131122xx104A−x44+02x1−1122111B==A21104+x−xx+1x5xss1B4−12x121040422x−+4x=3+−−11x1133142x15+14=221s22x+−−4x151+xs3311222211+31AA11322x72AA31x721+411750311x5333−+BBBBx7231−1x1100444−+122xx5335xx2−4~~5x22R1xR221x1xxx0xx4−4=34112213++−−−−44~~~~~~==x104=1155x100100x0=x1−+==R=x==x114=22=R1044xx2471xx30x3110000x1111=331110000x115RRxx1003377535310072=1=3122+−1x−3=x7722=33−1122−151007−01++−−++12=115+=−−17=−−21132222−−3xr1110000511111=R−=1441212001rrx3721R533113xr11~~172~~3rr1xx−1111x37−−53xx+−+0002110044−−7442−3−1002244++1+11−−−+44−11==42r1111110000−110000==0312==3311ss~−~rx−−412r−x1100311550ss5577~−4~−−r111x4+−x00440044−−=110000−−−1004100=334+11113=114431−4s4=1110044100=0044=s5−370s573140−−s5−100−711111002243311705−−5771122−1001121111511775005−111RR15−−11−2551530222−−−R55−R1RR00RR441102−R+−−−R33113111331144113R44R12−R+−−++RR770055−RRR−0242−333−−−1R1R−RRRR2223551404322−−−−22225524R−−23−71054RRRRR33413RR705−111133RRRR5112−1−−11RR++15RR15RR2533RRRR22R13R−−222213−−−R+R33−R+3RRRRR2RR3−11RR222−11−232−RR31R1R11
x= r= A B

r xx33
x

การหา ินเ รส์ ใชเ้ มทรกิ ซ์

การหา ินเ รส์ ใช้การ าเนนิ การตามแ

2 1 0 1
 1 11 1 1
111 1 A= 3 AA11=A==323232111111 3~ 3 30 3
2 A −1AA~A1 131− R232 −
1 −023
ddeetdtAeAt =A=3=3−3−2−2=2=1=1 1 0 3
~ 1
   1A 2I11=2==2323231011111011101010101010 113~−13102 013 1 0
0 3
1
 = 3 I 210 =AA10I23AI2 31−32R2 3
2
~113~~121212103113~1131031121031010310113101310R13113R130R1R1 110
~ 1 1 1 10~−1210 −1031−12R1 −−
2 3 R1 B
2B~B0I
1 ~1133~~101010−1133~13113332−113310−131332−10321311333210R10210−R−R132232R2R−21−210−2RR21R1R12 − 2R1  I2I 2
~
A−1 = −A12−1 =−31−12 − 1
0 
 3 

~ 1 ~~10113~~~~110011001100−−10113~~1131011322−−10113−110−101312132−−−0322131312102113−0333−0331RR−03132113−−3−RR113RR3221312RR12−R12−−032133−113R13R32RRR2 212 − 1 R2
0 3
1

~ 0

222 2 A= 1 211132122 3233 3 A= 1 2 3 AAA−A1 A A
2122 2 0 1 2
0AAA==1=A002=021011 1011 1 0 0 1

0 00010000

    AI3AAAI=Id3dI3eAA3eAA100t~~=t~~=AII~~A=II33d=3113000011d00021=100001e00d~~==e1110000~~100t==1etAt111230000A1d021100111000000111002110000A1001e001=110000102=t=11100A−1321110211001000−1011222311000011021021=21001−132100100==1112100100100~~1110010010011100123100−110032−123A13211200121−100110000111100100−11000101I=−10−1002=100−310120=110−2211100100001001120−~~11000=1100100210110000100=−−2−1001110d10011100−−11100001200210012100112e100−10011001002t10R−−R2AR1110R110000R1021100R2R210012R1001=1002R−11100−−2−+R1−1−R+21100−100123−1002R12212+RR2221100R1R001R−−21R32RR1−0R1003+R2−3R22R31013222132−R1R001−R−10=+−322−23102111R1002R1002R323−RR11100R1221−−+RR22R1RR2R1−32−3+22RRR323
    II3I3B3BBII33BB I3 B I3 B
111 −1−−222− 21111 
AAA−−11−=1A==−100=0 0111 1−−−222A−−21 = 1 −2 1
0000 01111  0 1 − 2
000 0 0 1 

 −  0 
1 1 1
−~110000 −A14A1A~014100−−−0131−−−−−14131116 0 01
1 A−−A0A12===116−−116011613−−−−0−0−1201212 000 1 − 1
A = 1 −−−111 A~ −−1013−−−610
3333 −−−333
6
−−110011−1−−1010012 0
~100001 0 −11−10 0
ddedetetAtAA===000+++666+++000−−−000−−−222−−−333===111 ~ 1 1 −11 − 10
0 ~ −1−4−2 21
det A = 0 + 6 + 0 − 01−112 −−−3−11=11 000111 000 000 0 −2 −

 AI3=A116AAII3I−3−0312===16−−1601613000 1−−−−1−00−131313000010000111 000 00
−−−2220 000 111 I3I~B3 ~B10000I1003 ~10100100−−−10 20 −0−3−3 322111−−
31 −−10−34−−2
0 B 20

1 −11~~~−10010110 −−−1111 0000 1110 000 000
~ 0 111− 1−−−1111−−−1011 11R12 0−00R1 RRR222−−−RRR111

0 4 −0030 444− 6−−−3303−−−6166 0R003 −1161R 1RRR333−−−666RRR111 − 2 − 3− 21 − 3 1
 =− 3− 31 1
1 0 −1111 10010−−−10010−10010−232012−−00−−0011−−00−−−11−1−11−−−1143311−−−−−324−032−0−122301211112100−−−−−−−RR−331133−111R2433114R+34+11111−R+R11100100433100RRRRRR2RR212R12RR12R++R3++3R1+1+3−RR1−+RR+−RR33+4334R3R3R4RR2R22222 A −1 =  −A3−1 −3 1
~ 0 1 ~~−0~11000000
0 ~~10~101010 − 2 − 4− 21 − 4
0
0
1 1
~ 0 0

0 000 000 111−−−222 −−−444 111

I3 B

 7 − 3 − 3
4  7 −A3= −−31 1 
0  A −1

4 A = −1 1 −01 0 1  A −1

−1 0det A1 = 7 + 0 + 0 − 3 − 0 − 3 = 1

det A = 7 +0 7+ 0 −−33− 0 −− 33 = 1 1 0 0
7A I3−3= −13 13333411011110310100−−−100−−−176666−132332106103100101−00101001RR100 R110R3211121++100+−00RR1766666R11160R3101−002101001ARR10I−1R3R32~=~21++B+−RR1111001006R11 R33341001002 AI−334−−110031~11=~ABI−11113100100A~=~111BI−13133400100100111100~=~100B433331403111100−−33410010100110011433334−−1003341004−−100RR1111112111334−−R100++100R1131RR343−111−331031003RR34311134331200−−1034RR1133433412103−−R++4RRR3114331RR12−−−
A I 3  =
~ − 1 1 −01
~ − 1
~ 0  11
1 3 ~−−3 1
− 1 1 0−1
− 1 0 11
33200
1 3 ~−
0 4
0 3 −
−
1 ~−− 1310
0 3
0 1
3 − 20



x+ y−z =1
2x − 2y + 2z = 6
x − y − z = −5

− x + 3y = −4

x − 2y + 3z = 9

2x − 5y + 5z = 17

A = 2 1 A −1
3 2 A −1

1 0 1 สง ศุกร์ ก นบา มง
A = 2 0 1

0 1 1

1. x + 2y = 6 2. 2y = 1

3x + 4y = 4 3x + 4y = 2

1. x + y − z = 1 2. x − 3z = −2

2x − 2y + 2z = 6 3x + y − 2z = 5
x − y − z = −5 2x + 2y + z = 4

x+ y−z =1 สง น ง าร ก นบา มง
2x − 2y + 2z = 6
x − y − z = −5 A −1

− x + 3y = −4 A −1

x − 2y + 3z = 9

2x − 5y + 5z = 17

A = 2 1
3 2

1 0 1
A = 2 0 1

0 1 1