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MA 2 2 6
ihlU 1.1.1 :
2 MA 226
r aI2 . . . aIn 1aI2 . . . aIn
A, =A, =
+di?ailJ 1.1.3 : Az
MA 226
~a,“?eh 1.1.4
A =
aI, = 1, aI2 = 0, aI3 = 4
azl = 2 , al2 = - 5 , a2) = 9
A = [aid tdkl i = 1, L,...,m
j = 1,2,...,n
A = Ii -: 1 :)jnh = (-: ; :)
MA 226
iYaiaof.ha 1.1.6 : 'It? A =
Tr(A) = 1+5+9 = 15
5
x2 = 2 y2 = 0
1.2.2 nmmudnr’
(Addition of Matrices)
u’u1u 1.2.2 : 81 A = [ad URt B = [bij]
~&wflumuo4raminml A LL53X B 9XLM’7nyu
A+B = C = [Cij] Id Cij = aij+ bij ~50 i = 1,2 ,..., m URZ j = 1,2 ,..., n
&I.LA+B = c =
Wh<l c,, = a,,+b,, = 3+1 = 4
Cl2 = a12+b,, = 3+0 = 3
Cl3 = a13+b,, = (-.l)+(-2) = - 3
c21 = a2,+b2, = 7+3 = 10
MA 226
Cl2 = a22 +b22 = 3+1 = 3
c23 = =23+h = 3+4 = 7
n’?Oh 1.2.3 : 0 0 0i
0 0 002.3 =
0 0 10’03.2 = [ 0 00 0 0 10
lIU1WlIQJ A ttfWh~iiA+O = ‘A t5l’iIctiuntuvl~n~ 0 hiju identity ilnhllmJ??l
tw~lttijutUQlj:n~~dtilr~Yann’ytuoljn~’dutthru’rii18u~i.n~~u 3 td&uhl tti3r
61 A+ B = 0 t&a L3l5t~ndla’htUvl~n~ B tihht?oia (inverse) Fh?h77UaTllIoJ
tuafn4 A ttficocl~ikpTnw& -.4 tu’uuttnutuol3na’ B bi72~
A+B = 0 && A+(-A) = 0
&Oh 1.2.4 :
1% A = [ -: : -:I
El - 1 - 3 2B = -A =
2 -4 -3 1
&&~A+B = A+(-A) = 0
MA 226
(Subtraction of Matrices)
w”UlU 1.2.4 :
A - B = A + ( - B )
A - B = AI--B) =
8 MA 226
ifad 1.2.6 : t$ B =
38 =
7Lxhln ( - l ) A = - A
1.2.5 "lJ~m~Lw~19601n~n~
(The Multiplication of Matrices)
~t3U.l 1.2.6 : t?'-; ? = [a,j] 6hlWt~Tlflf~Wl mxn LLAZ B = (b,i] lif)~UJVl~fVa
llul@l n xp wk3~KIAuoWJpl~n~ A LLR: B &I AB = C = [ci,l hk
c,, =. i a- bk, =t=, Ik a,,b,,+aizb2,+...+a,,b,j
id8 i = 1,2,. .,m Lb% j = I,2 ,..., p
MA 226 9
aIt a,, . . . aIn
a2] a22 . . a2”
ait ai . . . ai,
amI am2 . . . amn
b btzII s , . b,jr b -... tp
b... 2p
b... np
=
Cl1 Cl2 . . . c1p
C2I c22 . . . czp
cl.cij
Glt Gil2 . . ’ cmp
‘J:L~U~-l21U-lFIVOJ A 60 2 x 3 LtRfVUlWdOJ B ?%I 3 x 2 &hWMQnt AB Ml\Zi
tY+~ltilU?UWl6Jud%DY A = iW?Uttn?lIEN B = 3 ii?‘LziC 6OURQTttWNL%JV6ni ALA
UR: B UUclO AB = C 6&UlWOJ C 6% 2 x 2 UR:
c = [Cij] =Cl1 Cl2i 1 IU;tu’i = 1.2 LWj = 1,2Cl1 c22
CII = &aIkbkl = a,Ib,,+a,2b2,+a,lbsl = (2)(3)+(-1)(l)+(3)(2) = II
~12 = ,$,atkbk2 = attbt2+at2b22+a,3b32 = (2) (0)+(-l) (4)+(3)(2) = 2
3C21 = k;,a2kbkl = a2,btt +az2bzt +a2h = (4) (3)+(2) U)+(O) (2) = 14
1 0
.~2 = ,~,adk = a2tbt2+a22b22+a23b32 = (4) (0) +a (4)+(o) (2) = 8
MA 226
6Jitl.i A B =2 -1 3
4 2 0
'lun'luoJlQo7n'uo=riiu~:l~
3 0
B A = 1 4.[ 1 2
42 2
eYai?bh 1.2.8 :
MA 226 1 1
ocrAui-I AB = 0 h&i A # 0 LLR:: B f 0
12 MA 226
iih.il~ 1 . 2 . 1 0 :
1s1 0 - 2
A = L 1 tiluruo13nB4Jijvul” 2 x 33 1 1
&IL A’ = [i] 5l~iJllU7RtilU 3 X2 LLRt aij = ajiLauuao a:, = alI = 1, a;, = aI2 = 0, a\, = al3 = - 2
a:* = azl = 3, a:, = a22 = 1, a\, = aI3 = 1
A’ =
MA 2261 3
2. I< A = [: : :], B = [; ;I, C=
(1) C + E (2) 2C - 3E (3) AB UA: BA
(4) CB+D (5) 3(2A) UN: 6A (6) AB+D2 UA: D2 = DD
(7) A(BD) UA: (AB)D (8) A(C + E) LLR: AC + AE (9) 3A + 2A LW SA
(IO) A’ (11) (A’)’ (12) WV’
(13) B’A’ (14) (C+E)’ (IS) C’+E’
14 MA 226
(1) (AfB? # A2+2AB+B2
(2) (A+B)(A-B) j, AZ-B2
10. tb c tih.mnairl~ 9 tttx A, B tilutu~jnd?‘q~~kuulntall?n’u w~pii~
(1) Tr(cA) = cTr(A)
(2) Tr(A + B) = Tr(A) + Tr(B)
(3) Tr(AB) = Tr(BA)
MA 226 15
1.3 a~sru~uoJfll~nr=n’l~~~~l~6~~~~~
Algebraic Properties of Matrix Operations
&eh 1.3.1:
I12 31I -2 1 4 j + I: -: -:1 = [: : :I = i: -1 -:j + [-: : :]
Yltplij 1.3.2 : sl”l A, B LLA:: C r%4tlJQljn6W4ln m x n L&J
A+(B+C) = (AtB)+C
c
f&JOoa’: I w ” A = /a,,], B = /bi,j blRt C = [q,]
Gi”Jh4 A f (B + Cl = [a,, + (b,, + c,,)]
= [h, + b,,) + c,]
= (A-tR)+C
1 6 MA 226
hh 1.3.2
I
1 2
1 (F
2+
3 4 3
(F
12 2
I I+
34 3
&&.t
i
1 2
1 (1
2+
3 4 -3
1 3+
_. 2 1 L - 2
1 3+
__ 2 I> [ -2 :I = [Ii I+[-:
2
I=
1
1
I=
1 %
1
- 2 11
i 6 4 4 3 Ii 6 4 4 3 1
MA 226 1 7
ciative Law for Multiplication)
nqnij 1.3.4: IK r LLt% s tilu~tnm%I @J IfiA tihJol~n~lIulcl m x n LLA: B
tilutu~fnQuuw n x p 1:IG
(1) r(sA) = (rs)A = s(rA)
(2) A(rB) = r(AB)
GQQd:‘IK A = [aid LLAZ B = [bij]
(1) r(sA) = [r(Saij)] = [da,,)] = (rs)A
= [s(raij)] = s(rA)
(2) rB = [rbij]
A(rB) =
= r(AB)
thidl~ 1.3.3 :
b? A = 4 2 3 LLW: B =L 12 -3 4
WA) = 2l’: _; ,I] = [:: _ ;; :I] = 6[; -; :]= (2x3)A
18 HA 226
A(2B) =
2(AB) =
&&t A(2B) = 2(AB)
mqtdi 13.6 : (1) ni A tttc B ri%uuaJna’ww mxn ttix C tilutumfnGuu?n
n x p tth ( A + B ) C = A C + B C
(2) 0’7 C thtMI~VhU1~ mxn LLAZ A, B thtUR%lhl~
n x p U&l C(.A+B) = C A + C B
I< A = [aik], B = [bi,] UAZ C = [Ckj]
6&t A + B = [a,+ bi*]
(A+B)C = [ ,t,(as. + bir)ckj]
= A C + B C
MA 226 1 9
c;tiedis 1.3.4 :
IG; A = [I -: :], B = [I ; -:] ttA:: C = [;
(A+B)C =
1 o-
AC-FBC = 2 2
3 - l -
= i’: J+ [: -:][ I[
1 o-0 0 1
+ 2 22 3 - 1
3 - I -
-18 0= i I12 3
0
2 &&t
- 1 I
&lk4 ( A + B)C = AC-k BC
omnt-p~ 1.3.5 r5india-h ni~uanuA::n,la~nrtua~n~‘rilurlJ~,lungnls
nX?lU (Distributive Law)
nqrr; 1.3.8 : fil r, s Shbw~naGI~ “1 uf+r A, B Jhkhhm m X” t&-J
(1) (r+s)A = rA+sA
(2) r(A+B) = rA+rB
GpOd (1) 1% A = [aij] &l&4
(r+ s)A = [(r + s)aij]
= [raij + sad
= rA+sA
(2) 1% A = [aij] LLWE B = [bij] key
r(A+ B) = r[aij+ bij]
2 0 MA 226
= [r(a,j + bij)]= [raij + rby]
= [raij] + [rbi,]
= rA+rB
?lqMjj 1 . 5 7 : $1 A ’ LLR: B ’ &I ~~1~~hWll~JLaJ’hl~ A LLR: B 81lUh+h
6rtxnS1 c iionmai3sl 7 &&A
(1) (A’)’ = .4
(2) (A+B$ = A’tB’
(3) (cA)t = CA’
(4) (AB)’ = B’A’
MA 226 2 1
& (AB)' =I1 7[ 113 6
A' = [I ;] UR:: B' = [: ", :]
B'A' = [; ; ;] [I ;] = [;: ;] = W)'
2 2 MA 226
1. P4llt%Y-h A ( B C ) = ( A B ) C lib
A = [: -:I, B = [ -: -: :] tt8: c =
2 . WllafNil A(rB) = r(AB) lb
A = [; -;], B = [-; -; -:] 11% r = -3
3. WllWWh C(A+ B) = CA+CB ld0
A = [; 1: 21, B = [; : -;j LlR: C = [_: -11
2 - 3
4. ilJll1619’h (r + s)A = rA+sA lb r = 4, s = - 2 LLRE A = i4 2 15 . il~llffFlJ~l r(A+ B ) = rA+ rB l;O
A = [; -I], B = [ -! i] 1lR: r = -3
6 . ?JllWlJil (ATB)’ = A’kB’ LLR: (CA)’ = CA’ l&
A =
7. ~JllR@lJil (AB)’ = B’A’ lib
3 -1-
A = ttat 8 = 2 4
1 2-
LlAZ c = -4
8. PJMllWl%la’ A 11% B iJhl41" 2x2 LlR:: A # B # 0 11APh~~ A B = 0
9. WllaJ~h~ A # 0 ~JihJUl61 2x 2 ll&l~~ A* = 0
10. W4?18J&l~ A, B 11% C &hl" 2x2 LlR~ril~ii AB = AC h”i B # C
11% A # 0
MA 226 ‘23
&llU 1.4.1 : tYl~:t%JfN%JtolffPl? A 'h dU symmetric 61 A = A’
(sv?tih 1.4.1:
A = tih symmetric matrix
LW(flrh A = A’
I %llU 1.4.2: t3TJ~t~~llt%Jfl~fl~ A iI skew-symmetric 61 A = -A’
i?d"aeh 1.4.2:
I< A =
i
o-4 0 3
4 0 l-2
o-1 0 5
-3 2-5 0
2 4 MA 226
A’ rz
-
0
- 3
1 2
0 - 5
5I
0
0
1 - 2
3 I
= A0 5
5 0
nqnij 1.4.1 : ii? A t9u6rJ~Sna’~q~~l~ 7 Lm1ulx-lAuu A 1JiotjlupJ
!4JAU?flllOJ symmetric matrix LtR: skew-symmetric matrix 16 ttR~P3fh.4MtGlJ
tt~?Jt~U?LV’h~U i%.&O A = S+ K T’Nd S <O symmetric matrix LLfC K %I
skew-symmetric matrix
fiql$ ,S = :(A+ A’) ttt% K = :(A - A’) 'W%4h S th symmetric LLt% K
tau skew-symmetric &<LW~lZ’h
s' = [!-(*+A*)]' = ;(~+A31 = $A'+A) = s
LtA: -K’ = -[ $+A’)]’ = - ;(A-& = - ;(A’-&
zz $A-A’) = K
MA 2 2 6 2 5
th skew-symmetric &tdO~d W&lJw^~?6’h A L&U~t%l~?UJd S + K %fLw’UJ
tt.l.Wt&j? hlWJJukh A Rlul~ntQUU’l~~~~UJ~~~~~nYOJ symmetric matrix
LLAE skew-symmetric matrix 1% flttlJLlMdJb% A = S1 + KI I@lUd S, L&i symmetric
LLA: K &I skew-symmetric LflTiZdhJ~$‘h S+K = S, +K, U’UtO %&lJ
fiQ'lii1 S = S, URE K = K,
‘3111 A = S,+K,
+YJ& AL = (S,+K,)' = S;+K;
= S,-K,
t?‘dTlE’h S1 till.4 symmetric <JI.?U S: = S, LLAt: K, th skew-symmetric 6JlfUK: = -K, tts3:
A+A’ = (S,+K,) + (f&-K,) = 2S,
tmr
;;Ju”U
PY-meilS 1.4.3:
14,
tmt
i)fthh
s, = ~(A+A') = s
A-A' = (S,+K,) - (S,-K,) = 2K,
K, = ~(A-A') = K
1 712 3/2
s = +(A+A') = 712 6 3/2
312 3/2 3
K =
A = S+K
2 6 MA 226
GQlU 1.4.4: 61 A = [aij]bb&conjugate 9109 A GO A = [ail] dQ aIj ;O conju-
gate 9104 aij
cYaix.i~s 1.4.4 :
M A = [,ii :,:,I &ispdu
A = [ z_ii 13::]iTat& 1.4.5 :
12 1
Iii A : = [03-12 1 -3 1MA 226 2 7
t!tLhrjl fiUl~ll~nor?lloJ A lihilU?W?J LLRt: conjugate ?l0Jh?U~fJ?%Q
Vh?u”ULOJ LdU 2 = 2+0 ,i 03iU<l conjugate 110J 2 &I 2 = 2 - 0 i = 2 ‘&lfU A = A
?lqMa ,1.4.2: A Lh real matrix 7%0&l A = A
&a&s 1.4.13 :
MA = [2:,i :iZIJ
ihlal 1.4.5 : ~TKCzL%JfW~~rl~ A h&4 Hermitian 61 A = A*
&dlJ 1.4.7 : A = Lfh Hermitian matrix
z28 MA 226
ihlar 1.4.6: t~l5I:l~~fllXJol~fl~ A h!h idempotent 81 A2 = A
hWil4 1.4.8: fil A =
A* = [: “oi[: ;] = [; “bl = A
$.$I4 A b%4 idempotent matrix
&Jlal 1.4.7: 1,51'%L%fW@l~fla A h&4 nilpotent 61 A” = 0 L&l k tifll.4
~lUX.4L&.JY?fl’~~ ‘j Ut3SItL%Jn A 41 nilpotent of index k t% A’ = 0 tt@!
A’-’ # 0
tih.h 1.4.9: $1 A =
A* = [-: -:][-: -:] = [I ;] I=OLLREA#O
gaI?U A L%-4 nilpotent of index 2
$.llU 1.4.8 : ifI A I,hLoJ~~n~~~~~ LTliltLifJll A 'hh diagonal matrix
MA 226 2 9
9Yadl~ 1.4.10 :
D, =
?& D, LLFiE D, lh diagonal matrix t~lil~t%lU~Lhh 7 ~lhlOJ diagonal matrix
&ii
ihl 1.4.9 : n'7 A Lh diagonal matrix L7lll:L~Un A hhRLflRl~L%J~fd
(scalar matrix) hiatnal~~aduwuu?t~unuuJyuuoJ A LYiln’U
iYaiaah 1.4.11 :
2
L 0
0 0
s = 2 0 tih.tatnfiliruvSni
0 0 2
1
C
0SC L. : 0 . . . 0
c 1.. 0L&I c riluatnalrl~ 3
0
1o...c
3 0 MA 226
Sijflh 1.4.12 :
I hoi 1.4.10 : Identity matrix #“SO unit matrix ?%%3tUfldUJ~%&J?J c = 1 I
I” =
hH.il-a 1.4.1s :
MA 226 3 1
= AI, = A
ihol 1.4.11: tmct’iunumfni A 'il involulo:y 51 AZ = I
Lillu upper triangular matrix
32 MA 226
1 0 0B = [ -1 3 0 1 t&4 lower triangular matrix
1 0 - 5
MA 226 33
1. F)J~@dl hJol~n~ A th symmetric U&9 A’ Odh symmetric dhU
2 . ni A tilutu~fn&@% 5GpZ7
(1) AA’ LW A’A t&.4 symmetric
(2) A + A’ LOU symmetric
(3) A-A’ th skew - symmetric
3. IhJoli’n~ A LW B th symmetric WGigT~tiil
(1) A + B t6U symmetric
(2) AB l&t symmetric l%ltdO AB = BA
4. P~t!hMJol~n~ A \%$UJLhWlJXlllO~ symmetric matrix LLR: skew-symmetric
matrix t&I
3-2 1
A =
I I
5 2 3
-1 6 2
(1) (CA)* = EA* (2) A = x
(3) (A)’ = A’ (4) (A*)* = A
( 5 ) (A+B)* = A*+B* (6) iiii = iiE
( 7 ) (AB)’ = B*A*
6. #I A t%.WJ~‘rfdhl 7 %$$a7 A*A LLA: AA* th Hermitian
7. fhJol?n~ H dU Hermitian PJ$fil~~l B*HB Il:th Hermitian &Xl
8. ~~~~~d~lR~l~n~~~‘lUtt~~t~~~tt~J~~~oJ Hermitian matrix thIh?UO?J
9 . 51 A tilUtuAna’Cq?~~J A = -A* (iiU&~ aij = -3 t~icxiuntu~3nB A +h
skew Hermitian PJw^~d+h
(1) iA ti)U skew-Hermitian $1 A th Hermitian
(2) iA th Hermitian 51 A tfh4 skew-Hermitian
3 4 MA 226
IO. $1 H th Hermitian LLA:: H = A+iB t&l A LLRt B t&.4 real matrix OJfi@'h A
th symmetric LLRZ B th skew-symmetric
11. ~J~ltJJ~~Vh4l@l 2 x2 ttAtti)U idempotent $Ja 1 O~h@lJh~lLI
12. %lJMl diagonal matrix $JhUlW 3 x 3 ttA:Lh idempotent
13. F)JMlLUBI~&‘LlUl@ 2x2 LLRtdU involutory
14. Qln (A- I)(A + I) = 0 +dPIHiQbi-h A = I V&l A = -I ilJWlV=hQrilJl.h~fWl
15. ilJLLRClJ~~t&.&~l A + B LLRZ AB t&.4 upper triangular 61
S” =
MA 226 3 5
,A2 = [h i]
A =
alI aI2 aI3 ! aI aIs
azl a22 a23 i az4 az5--__--__-__-______ f -____-___-_
a3] a32 a33 (1 a34 a3
3 6
A =
MA 226
A,1 A12 Al3f&l A22 A23 1
51 B =
MA 226 3 7
I$‘= ------A- - _LLRt: B = . . . _..-_.-.--- + .-_-.-- -
A,, + ‘31, &2+ 4,A + B =
A21 + B2, A22 + B22 1
38 MA 2 2 6
&& AB =A,,Jh, +A,,&, AIIBII + A&,,
.&,BII +Adh &nBn + A&2 1
9Yah 1.6.7 :
A = LtRt B =
MA 226 3 9
AB =
=
h+[:l i: :I+[: :I Liz :I+[: :I1[o] + [o] [o 01 + [o 01 lo 01 + [o 01
[21 + [o] [o o] + [o o] 10 01 + [o 01
4:o oi2 0I
610 OiO 0--A---;---- =010 o:o 0
--* - - - - + --_-2 : o 0;o 0- :
a
4 0 0 2 0
6 0 0 0 0
0 0 0 02 0 0 0 0_ 0 I
40 MA 2 2 6
6 3 5 1A B =
2 7 - 4 3 1
2. WHlUR~fU ABOi A = [ I20 0 1A,LLR: B = B, 0 B,
0 I, B 3 1do 12 = I 1 0 1 , A , = [3],B, =
0 1.
alI aI2 aI3A = [ 1azL a22 a23
a31 a32 a33
MA 226 41
r 1 0 oi 0olo 0 f : ’0 1 0; 10 0 1 ; 0 0; 1---------.-.-------L--------------~----0 0 o! 1 0; 0
0 0 o!o I i 0--_--__-__-_--_-__- f _-_--_--_-_-L_ I___-1 1 1:o 0 : 1,
4 2 MA 226
GJlol 1 . 6 . 1 : ‘lpll’ A t%.uu6in&iul~ n x n ibitWl%l~ E $J?'hl$:
AB = EA = I, ttt% taiP~ndl?-h B th~Ut+li~IiOJtWl%l~ A tt%33%lU
ttw&iai?u A-’ &&I B = A-’ GiJh AA-’ = A-IA = I, n”7tuhviT A Ght-daia
L511:fdl12h A tfh nonsingular ttd?% A r~J~ettaaiat~lc=ndla~~ A dU
singular
A = [: :I B = [ -: -:I‘QJ=tV?Wil AB = BA = 1 GJ$U A th^oUt3~i~Y~J B LLR: E th~Ut?&WlJ A
A =
b.bnirKiutaoinuo4 A& t31auuGi~1 E =a b
c d
MA 226 4 3
nioi
a+2c 2;4;] = [; ;] &iJuu2a+4c
a+2c = 1 b+2d = 0UR:.
2a + 4c = 0 2b f 4d = 1
I I
w4gp6 : UUlJ6ibbJ@l%l~ BLLWZ CLih?hL?oiRuo~ A&u
AB = BA = I LLR'. AC = CA = I
tdu B = BI = B(AC) = (BA)C = IC = C
cs”3Qdls 1.8.3 :
A =
I$ A-’ = a b[ 1 Ot&iOJMl a,b,c, d8&11flc d
AA-’ = A“A = 1
4 4 MA 226
F1lt-1 (1.6.1) FE\&
....... (1.6.1)
....... (1.6.2)
a+2c = 1 b+2d = 0LLAE
3a+4c = 0 3b+4d = 1
u~~un?~~~~asst~~~~~~~ a = - 2 , b = 1 , c = 3/2 LEG d = -l/2
q1f-j (1.6.2) ??$I:% a = -2, b = 1, c = 3/2 LLA: d = - l/2 LliU&J?n’u
*-’ = [ ,: i2]
iTat& 1.6.4 :
?l$M~ 1.6.2 : 87 A LLAt B Vil\rn”lih nonsingular 11&a AB WilU nonsingular
LLA:: (AB)-’ = B-IA-’
MA 226 4 5
iiigo; (A@ (B-IA-‘) = A(j@ = ,&IA-’ = A,&-’ = 1
(B-IA-‘) (AB) = B-‘(A-‘A)B = B-‘IB = B-‘B = I
6Jlki (AB) ( B - I A - ‘ ) = (B-IA-‘) ( A B ) = I
u”UiiO B-IA-’ tih?ht?E6tW1J A B %&I (AB)-’ = B-IA-’
A = [; ;] t B = [: ;] ; AB = [,; ,:]
A-’ = [ -;,, ,y3] , B-* = [ -;;I -:;;I
(AB)-’ = [- 13124 5124
10124 -2124 IB-LA-' = [-:;: -:;:][w:,3 ,e,] = [-:2 -:y = CAB)-’
46. MA 226
(3) r_ 1 1 00 1 10 0 1 1
(2)
(4) [ 1 0 (5)0 1 0a b
0 11
1 0
a 1
2 . wJ4plil (*-‘;-I = A
3. WJi+l (A’)-’ = (A-‘)’
7. w4wm5nG 2 tuvhd ~JiillU7@ 2x 2 LtAdlU singular tt6iwauantih nonsingular
8 . ~JnltSJ@l~?la’2 LU41?&!JhlWWi 2x2 t6ii’:th nonsingular tt@kXJ7dU singular
MA 226 47
4 8 MA 226
Kxlfh~ 1 . 7 . 1 :
18
61 e =
&lh e(A) =
3RI + R1
1 - 3 t
0 1 1
0 - 8 9
ii7IK e, i%nisnr%muttn~ (91lumGi) tmud 1
e2 &m75nxiimuttna (mufwcju6) ttld 2
e3 Gm-mrciimuttn~ (ollumCu6) ttu.4 3LA
uumi el = Ri++Rj (Ci ++ Cj)
e, = cRi (Cci)
e3 = CRi + R, (CCi + c,,
b%&
e;’ = Ri ++ Rj Cci ++ Cj)
e;’ = I/cRi (1 /CCi)
ej’ = -cRi+Rj (- ccj + Cj)
9:tGiAjl I-,, ez, e3 t8unl5n~cn’itt~ytiu~~~ e;‘, e;’ LLR: e;’ 61lUhii
%llU 1.7.2 : $1 A LLA: B t?hW.lol?n~li~l~ mx n t~l?:ndlT’h A row (column)
equivalent Tiil B 61 B b%lnnl5n~chlla.utl~ (011Wl0&~1.$ llUtu96fPii A
datiodn’uiclfdfil
MA 226
Pi,atiseilS 1.7.2 :
b%aiaefhs 1 .7.3 :
'317lklU 1.7.3 'ilLLh4<1 $1 A row equivalent M% column equivalent n’y B tttkA 0: equivalent ??IJ B
50
,
HA 226
(1) A row (column) equivalent n’u A
(2) 61 A row (column) equivalent 6J B l&J B ?Z row (column) equivalent kl A
(3) 61 A row (column) equivalent %I B LLR: B row (column) equivalent %I C l&l
A Pt row (column) equivalent 6.l C
2 . PJi?pz~l
(1) A equivalent l%J A
(2) <I A equivalent T%J B l&J B IIT. equivalent <Ll A
(3) 51 A equivalent fk B 1lRE B equivalent l% C l&I A ‘Jr equivalent n’u C
3 . F)JllR@lJ~l A 1lRZ B equivalent f6.4 <‘l
A =[:I j;]llRE B =[; -;;;I
4. smxtiwuru~^an4 A wnxn’lRuol3nQ B ni
(‘)A=[;-;;-j: B=[,;;;:
(2)A = [“‘I, B =[‘;;I
MA 226 51
1.8 nJn?na;Iasa#u
Elementary Matrix
PhdN 1.8.1:
E, = [:‘3 E2 = [; -%;y
E, = [;;I E4 = [i:]
1 0 3
ES = [ 10 2 0
0 0 1
52 MA 226
MA 2 2 6 5 3
nqu~ 1.8.2 : n”7 A LLFiZ B t%WJo1~flh&l~ m x n tth A 0: row equivalent
(column equivalent) n’u B 6hdD B = EkEk_,...EIE,A (B = AE,E,...E,..,E,)
t& E,, E2, . . . . E, t%ttU@jn(a’t~Q~6U
&pi: !$e LhTllarlatYh~n 9 45 e ( I ) = E
81 e-’ t~U~wtaQ~a~Qsnlana~~l e ttt% e-*(I) = Em’
hti4 EE-’ = e(E-‘) = e(e-: (I)) = I
LLA=. E-‘E = e-‘(E) = em’(e(I)) = I
&lb4 EE-’ = E-‘E = 1
i&i0 E-l G-htaoimtos E ttart&wain e ttn:: e-' t8~ni5nat~ittzlutiio?n’u
&lkt E tm: E-‘’ t~Pttuajna’tdoj~~tt~~t~~~~~
5 4 MA 226
MA 226 55
echelon
0 0 0 0 2
0 0 --I 0 0
0 0 0 0 1
0 0I
OQO
56 MA 226
vYa”aedis 1.93 :
1 0 0 0
0 1 0 0 , B =
A = [ I0 0 1 0
0 0 0 1
12001 .1 10 0 1 2 3
0 0 0 0 0
MA 226 5 7
c = 0001 I -2
0 0 0 0 ' 0 0
- 1 0 0 0 - 2 4 -
0100 1 8
0000 0 0
+ 4
VlqXI~ 1.9.1 : tm%duuw m xn IFI 7 9:: row (column) equivalent n’YtumSni
row (column) echelon LBUO
58 MA 226
&di 3 w^9lxultuolln~ c = [cij] t& c ihm3nBdoouo4 A hvhtna
ttmoon &A4 c ihtlar~u ( m - l)xn n’&ttd$U4 1 ou&#i46 3 n’Yrmfn4 C
row echelon 44 row equivalent %I A
ta.m~nim.m m x n 161 9 ‘Jz row (column) equivalent rihm%vii
row (column) reduced echelon MU0
RI ++ R2 0 0 0 2 1 =rhwQn~o~~uiittwPiJ (1,2)
0 1 0 1 2 llmu 1
-l/21R, + -R, 0 0 0 2 12
0 1 0 1 2-
MA 226 59
R, + (- 2/3)R,
0 0 0 2 1
!
0 1 3/2 1 -l/21Rj + -R3 00 1 0 -*5/3I =2 H &Of@@ row echelon
00 0 1 l/2 4 row equivalent 6l.l A
R, + (-3/2)R,+R,
R, .- R,- R,
dJO$WJd row reduce
echelon
6 0 MA 226
?lqEl~ 1.9.3 : ?l A &bW~~tl&l~l~ mxn I@ 7 U&?J A 8: equivalent n’y
w&Gb$u~lJ
i
Ik Ok.@-k)
o(m-k)xk %n-kb(n-k) 1
$sexiunii normal form
&xlha 1.9.5 :
1 01 0 1 2 - l1 2 - l
1 01 0 21 021 0A zzA zz
-1 0-1 0 -4 1 -2-4 1 -2
reduced echelon &i
R, + R,-R,
Rj + R,iR,
R4 +- R,-R,
1 0 1 2 - 1 ’
0 0 l - l 1
R4 +- R,-R, 0 0 -3 3 - 3
0 0 0 0 0
HA 226 61
0 0 1 - I 1R, + (- 1/3)R,
0 0 1 - l 1
000 0 0
0 0 1 - l 1R, + R,-R,
000 0 0
000 0 0
1 0 0 0 0
c,- (-3)C,+C, 0 1 0 -1 1
c, + 2c,+c, 0 0 0 0 0
0 0 0 0 0
c, + c,+c, 0 1 0 0 0
cs + q-c, 0 0 0 0 0
0 0 0 0 0
U 2x3
0 2*3 1
6 2 MA 226
MA 226 63
0 0 - l 2 3
02 3 4 5
03 2 4 1
row echelon LLAZ row equivalent I% A
(2) 9WWJW?lI~ C ~xI$I.&~~ row reduced echelon LLA’. row equivalent ?%I A
2 . rl 0 17
(1) wmsSnG B 45O$~~lJ column echelon LLRz column equivalent 7%~ A
6 4 MA 226
