2ม3-4-64

เมทริกซ์ (Matrix)

รเ ท รรรรเเ ททรรเเ ทรรทรเรรร2ทรม5มรร ร รรรรมม รรร222ร222ม055ม055334เิ ร433เิ500ร222005ม055 ิเิเ ิเเิ เิ
รเ ท 2205 2255
2250 2200 3355 3355 3
25 2255 3300 3300 3

4400 4400 4

 (())  (( )) (()) ( )


25 3343340500เ50เมมททร222ร222ิก055ิก055เซมซเ์มท์ 334ทร334222050เรกิ050222055มกิ ซ055ทซ์ ร2์22334ิก055334050ซ050222์ 222055334055050334334050050222222055055334334222050052022055055 222334055334050050334050 
22220505 

เมทรกิ2ซ5์
เมทรกิ ซ์

ทิ ม mn (m, n I + ) m (row) n
(colum)
ม กิ ท ij
a11 a12 ... a1n 1 )
a21 a22 ... a2n 2

 m

am1 am2 ... amn
1 2 ... n

aij ม ิก (entry) i j

เมทริกซ์ i 1,2,3,...,m j 1,2,3,...,n

mn mn (

m  n มิ ิ เมทรกิ ซ์ (dimension of matrix)

2102221121200221002020000121312112213113313 3333 222222

−3−−332−302221000111 1…111…3…333..
121122212 −−−−−11111 1…111…11…11..
3…333…11…11.. 22

…………………………………………………………2……2…2....2.

A,B,C,… a,b,c,…

A = a11 a12 
a21 
a21 

a, b

 A = aij mn A mn ij
j 1,2,3,...,n
aaij ij i 1,2,3,...,m

1 (((1(2121)1)())(11A)(((A((()111222aa111aa))))))(((131111(323(21(32(AAA)))112))1(aaaaaa)==1)(==1111333A)A222333A)aa1−1−aa1,3======A1,2133AA123AAAA,a,a111−−−==1==1a=4=,,,Aa4AAA111A2AAA231−,,,3aaa=1−=A,A−111A1,===A−aaaA3444=1603A=2,60a32222,3a3331,====a441,AAA=4a−−−42333,===a66600032,222a333=32−A=−2a−−−1−,,,Aa−13444−=603623=3136630423,,,(13aaa=43(23=,24222−−−,−−−)aaa4−−−111=)=,4a−33366644,333−a1114444(((2−333===−222a2−12a−−−2aa−−1421)))361423===11143624(34443−−−1=2444442(3,=2,)222aaa−−−=)=4444a222−1114111a4−2224444−2−223a−,,,32a−54321,153424,1241aaa24,a−−−222a,33333555333a4444,,,4a,−2,−32aaa3534,a533334,a44433a3,,,3a,3,43aaa4,a333,a33311a4,,,4a,3,33aaa3,a111a,44422a1,,,1a141aaa4,222,111

a
a2

22 2 2A =A a=ij 4aAij4 = 4 a ij 4  4

4 A = aij 4  4

aaaaa==−1−124311111010111aaaaaA−12341111ijaaaaa01111234=22=222 1

2 2a ij =a ij  i >0 ji > j i> j 1 i=j
==A j  i>j
aAij Ai−<1=ji <ajij 4 i4< a ij =  0
4
a1a2a1113 a1a31a214 aaaa41a2aaa344444124343333 a14  =  − 1 i<j
aaa234aaa222iiaa243aa111<>12343333 a 2a32a224 
a 3a33a234 4 a11 a12 a13 a14 
j a 4a3a4a21444 a24  a a22 a 23 
j a 34  a 21 a 32 a 33 a 24 
A 31 a 42 a 43 a 34 

a 44  4 a41 a 

 4 44 4  4

A = aaaa23111211 a=11aa22=22 aaa==aa=21aaaa323111a2212431a333a1==43=3==3=2a=aa1=aa=aa43332122343a234441a=a23=4=4==44=1a=aaa24=a=3a14444313a321=a2=4==4==2a=a1a2a=4a243424a41=a3=4=4==3a=aa31=a42aa4a43411−22211=01======−a0aa1aa3214342313 = a33 = a44 = 1 =
aa4112 aa==122aa1 a34a22=1=331 = a41 = a42 =
== a032 = a23 = a24

== a−141

11000=aaa1122111000==A−=− 1aa2−321 = 1a11110aa143−−332−1110=1100=−−−−−=11a1111aa234−−−444−−11111=11104==4a1a2444−−−2 =111=Aaa34=43 ==100 0 − 1 −1 − 1
0 10a=13 111=0=−− 1 4  4 −1 1 −1 − 1
A =A 0 10 00 0 0 − 1
A = 1 00− 0 1
0 0 1 4  4
−

1 − 1

3131))23313213)))))33a311331))2311)))))aa2323231))111)A3BBabb1311,)))))))))33311aaa1))22aa123111A1111Bbaaa11,111)))AAA3B1BB44aAbbb1111,,,2a1111,1,12222aaaa111111Aa0B4abA011111,1214444,,1AAA22,2a,2,,,1aa,0,1a,23baaa141000B223,Aab1A4B2a2111,2222,A4,,,222a2222,3,,,2223=a0baaa1,,,B3,a=222A43331bbb221111,BBB,aaa22AAA24444b,2222310a4=,2222b2222202233333,104b21=1==B02a,A4,,,1,2b,22104a,264bbb2302=1110004441a222264,20002222,b2,2,111122,,,,2bb2a,,,6422A10b−42B02aaa32b2666444222,bbb,12A1−21B,2222232,,,,222232222bb112222,1Aa−664B2bbbb332,,,AAA−−−2b11BBB,051,6233332222bbb2,11113111a,05b14,363333A−b2B2,111132a,6366,054b13,1,,b2,2,,0500b553a,32432bbb1336aaa22)4442b,,333332,222205322222,,,33333bb2)242aB2324bbb3BA322222)32=33222=,23243333=B)))2222BA086b224==B232=2222342244ABBBB164)=2=086BABABA=======0==86224164B000868686BA−164==0332=1611466442086−0332−1642240332−−−2030033323322A2222−4033224A2222444AAAA2B4 A B BB B
BB

ก รเท กั เมทรกิ ซ์

1.  b = bij mn mn
ท ิ ม1. 2
ทิ ม  abij=bbiijj mn i 1,2,3,...,m j 1,2,3,...,n
= j 1,2,3,...,n
 2A = aij mn mn
 A = B aij =1b)ij
2) A i B1,2,3,...,m
A = aij mn 1) AB
2)
A =AB= B AB
AB
A=B

1

1 11 ====A22b22a=ba,,1,,12112111=2=AA2==57bbaa7==2211,11,22,,22ba73==22==11221551A,77b77a==b,,a2,,=2111212772332=,,=B=2=2,,bb25aa27=75A2252211,,,11ba,===7132==1BB22252222,=2==b=55b2a55a52,,722,,717322122,2,22==55=B73=2bb32=aa35,225b7733a2222,,222221==12==5=33=B33b2a22=5732252,,=2=2533ba273222==33

a11 = 1
=
b11 2bb2aa,11,111111

ั ั ัั ัั ั ั A AA B BBA B B
ัั A

1 2 −1 1 2 w
=−−54541102,, 541122yy, =ww44zz2y 
2 AA == 102102 22 A − 3 22 B x 4 
−− 33 − 4BB −4
AA == BB −− 44 == xx z 
−− 44
ัั ัั
A=B w = −1, x = −3, y = 0, z=
z =5
ั ั ww == −−11,, wxx===−1−−,33,, x yy= ==−300,,, yzz == 550,

ww == −−11,, xx == −−33,, yy == 00,, zz == 55

((133))3(2(x(((((()3111333y))))))x−333((((0222((23x1xx3))),x))yyy()x3+13yxyxxxx+)x32(2yyx00333)xxxy,,2xxx(((((((xxx33322+++6x111133333=5yyyxxxyxxxy2y+++y))))x))xxxyyy2220yyyyy3x3324((,x(222x23+1xx3y2xy))222x6661+4)yy===55504x21yy2y21yyyyyy22xx5yyyyyyyy00xxyy222444200233xx26=5,,222y21x1x1y4y44x00((4422111yxx3y3y111++222115533000yyxxyyx0x00++))xx2422yyyy21404212212502222066==55yy22yyyyx---xyyyyyy---------2244------------221144---0044221111---22----55---0000---------------------------------------------(((---------231------------------)))------------------------------------------------((((((---231321------))))))------------------------------(((------132------------)))------------------------------------------------

444444 xxxxxx,,,,,,yyyyy zzzzz xxxxxx++++505005005+05yyyyyy −−−−−1111333331322222 −−−−−−−−−−−−3333344444yyyy4y ====== 005550052222052502 yyyy−−−−y−11111−−−−−1−22222zzzzzz xxxx−−−−x−−−−−−3333−344444zzzzzz

(1) y 2 x (5xxxxxxxxxx2yyyyy2222xxxxxx)xy2xzzzzz yyyyzzzzzzzzzzzzzzzyzzz (yyyyy2y222233333231111yyyyy1y1111x1) ---------------- (6)------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------(((((((((((((((((((((2213441355213514441325(((((51243)))))))))))))))))))))))))))))

(((((((((((11113333313))))))))))) ((((((((((22222(444442))))4))))))y) 2 x 1

(1) (1(1) ) y 2y2yxx 22 2xx x 1 -------------------------(--6---)--------(6(6) )

x3

x 3 y 2(6y)yx 22 x(x5) (5(5) )

(1(1)) (1(1()1)) yy 22yyyyxx 2222 xxxx --------------------------(-6(-6-)-)-----(6(6()6))
4 (ตอ่ )

yy 22yyyyxx 2222 x(xx5(x5)) (5(5()5))

22xx 222(x2(2xx2x xx)()((2(22211xxx)x))) 1111

22xx 222x2xx2x xx222211 xxxx 1111

xx x3xx3x 3333

xx x3xx3x (36(3363)) (6(6()6))

yy 22yyyy(( 23223)2) ((((55333)3))) 5555

xx x3xx3x (34(3343)) (4(4()4))

33 zz 3333 1z1zzz 1111

zz 22zzzz 2222

xx x3xx3x,, yy 33335,5,,yyyy 5555zz 22zzzz 2222

1.1. 2 3 4 5 2. 1 2 3
 
7 4 3
61. 22 3 4 5 1 3
25. 01 2 
3
3. 1 4 4. 3 7 87 41.
1 1 91
1. 1.   2.2. 6=38161024 2 5 89705a412213,0..a1248163,3a326241, 4 5 2. 1 2 3
23.. 3. 4.3.  
8 3. 64735 4 43. 73 7 a 25 7 4
1. 3
1. 2 −9 4 2.5−711 −42214 311 1 5 0
2. 337 b
3. A 3 1 4. 3 7 8 a

6 642b2816A ==c 6247010 − 3 61 59  1 1 9
0 4 3 7
a,81ba0c
2a. , b3. 74. 33 7 −81 13−29b.271 8 Aa=12,10a14−7,9a21−,3a3 25− 1 − 2
−2 9 14− 31 a976 9 6
3.1  9  a12 , a14 , a
5  3 43. 3 a,7b  b
3.
c
3.1baA22= −10b26Abaa072=2x−=c7=9b270−10264=−75323y570−=a7−96−62142525−−932aa3−6
3.2 3.2 c a a 3b.1b a 2 = 7 2 a
2b5 0 5 
a, b a a12 , a14 , a 210, a

3.1 a, b0 −1 00++−11932.a25353 a 2 −yy6a75−5−1112=,a24514 ,aa−6212, a25
26bxx
==b05052b

3.13.2 a 2 6a = b2245= 76  2 ax − y 8 = − 4 8
b 2 −70 a0− 2 5  3 22 2x + 3y
ba b 3

2