lnbr0 t0 LtneOr SglternQ
ts $enerot egn oT lrne in tR, qx -i by
* Senerct eqn e$ llhe rn tR3 qr. t b\, t cz = (,1
Lrne0r eg n lnn vc,r io bl eg )( , dz r' , i(n
clrXt 'l Qr- Xz { .. ..t
0n>(n = -/r'
ie
cons I an I
--D .rqrr obie occc.,r on tV 1() isi power
.-b {€lriotrles Onl g fylLri{ I ptl *, tr! c (Jr)SI crr:l g
*t€ *o knotlr, *
tf 3a-Qg=-l -> SOIU+lOng CGFresPqn6sa<i Io p Otn{q ct1 ttne
Es,uJ ( sts)- rl(r) =t5-rL=-r)
( r, rl ( str)-t, (r) .= J-rj =-,)
et JC:
L .. t."
v's
sb - ug t
er - [ i
I f.r ir *5 s- T
3
r tl =[
b
!r?
r.{ q rl rtl =r js - nj
L
J
x thes e qre Sat e{ lcht In ,[ a\u'l P or
q m elll c f Ol f-c')
,tY1 e tv\,e sglq o(e equ,vqlen+
x,SgglaY'n eI ttr\eor eQns -- {tnlle {01' s I llneOF eq ng
( o tt eqn$ n ov e J tl1l P vor icl bie\ )
g 5gtul i on of o .f q .t tefYr o{ llneqr eq n's = Ve c i€r 'ii.rcj[ tt 'J lff)ui lQn(('iiq
.s()ilri tcn a[ €Ocrr eq.1 rn Jq I tern
ra Sotut (O n set -- se{ oc (tlt eqla J Ctt.. ttfJr,(, n-i {r.;d I {-ii-t
x 'r 66lvrng *he JVI lqv\^ 'r =,. lit!16g( crt l,DCl,r\ LJ J(rrLr itL,, ' JQt
, )F O.fqStgm o[ \negr eqns wr{h reot caefftcienr( hoJ etther
" unlque {olu{tong \ aO.,Ur 6 | ent ,l qJ tefn,f
, ^,. .t' ^, )
^olrtL
porot$o''' -' ln I rfls t QltJ morrl'lO\uil('nl
o oo ,lolur lcnt Incongtslerrl sq6lenn
$sivin g ( JtJji0rTl Oi LineOr eqn{
r ti^,e ilnSOr Jq$rei1r( Ore equivCten1 rf rh€rj hova if\E
JC{fflQ $CiuiiLn [0iS.
.$ e qul\,qleni Ji;16 = $cii.) hova lJr1lQue JCI! il$r)l
+ APProQcn io Jotving
. t[onS t c i r']'l g\! E.- Sq,tiem ir)t0 eOJtet C,t-\e {o Jtivt
a flrrn icr irrongrrCrr pcrlrein
( ,l pcond eq u o'l lor) h o\S ono lotrl val I Qble -rhan (tr t r \
sr'f rnol ono
)T qn\.| JOlurlon cj+ giYon JgJterh rd 0lJc '"|Qlu'lorl
)e On\.I .rolltiOn oF finotsg,fl$m lJ 0l'f o JGtuilcnof grven6ns
3" $rven39r,n- qnd 'flnAl 'fqJ'erYl =QQUrvarenF
,Methods $or Jotvrn Lrneor e nS
'+- iY-r ,lJ r r- , . -- .. .3 l^, .1 G rl: -:l-"^- i-'- l^{'
M-fr,\ {-rn f ';.rt g1_pcr,-i1 il irr r{-
(, ,lt'ra1 r'!r\L (J'J frl t
COi )lt rgtrlrq trrliti,l 5 <r
L,l l.)Crl Ciil
tQodrnq I O 1rr i-!l .tl tluo lvr..'cii^{ , irr.jr rf in l(,ai +n,,.J , r.,(t./,,r.i
en\riQ' ) !ltli.J
[oYm f rs ln a co [u rn r) Jo +h$ tef I cf orig leqdlncj
5161r tolt \ er\1TrO( \r tiOir\ tl
pqHrErn
L+,n .ltrc1 cct\-r i1,\r-1
c_(rft\<.llr-1rr1 q q tecr <d er:rf u q ji1
, flit cf r
enlrl e:r t) rlio^, I ecxcirnS <nri(j €iie (!
)a- Ll tc. tr)tll t(i!\, I ci\ oF el g-trQr)q
a, ttlel\jt i!t1': lr;1..r Cctr\ b€. pet fc)rrn€d c,.., $\Jf l pmr Of itn 651r e5Qh$
.lC ri'dr1,i lC,r1'\ r; .tO eq Utvol en+ ,f'rJ.f l efn.
r\lpog L] .il ii {l '.i u! .: 1'Lt ,! .!
li^r I r.:r'
\+' (!l\'rtt1rty t i'n l)V tl!')l€tc !Crl5r(il)l
6- A.rcl l\l!l,ll) 1g vr I !!\ r\- !t)a,l\er
--. 1.,(.(a'A\ ti a, tr e ':J rCv\ i(iriu (l tC.-\ \reou(\. r-t)ulir\ itj s,ct)etci1 t(n,'
r i4Olrlces A Ond R Ore rOw equ,uoienl' if rhere
qr0 SOqUQnCq Ct ete fnen,Orq fOw C,petof ton( iherl
COnvsrlS A lo g
L+ tl tngg Or o f oW equrucrle.nF -> c-a r\ lce hedr,..rce d
i O {Afne row ech\€lc}t\ t oYrn .
x- G Qr..lg st en E ii i1.}1fiti l I C,1
@ writt 0uS f r'1 rll i ril nt;ir,x O r* dr.r('G 1e r Q!^, eciaei orr -scr rrt
@ trso hocr {(Jb {r ilu +l orr rO JQlva
*-t> rqnK o[ rrroit,x * of non aero ro \A,B ID {C,! e! la crl Cf\ i Cif).)
ec. f qn( tA) : rqntt of mottr\x d . ,-., .hr.''
.-'\ n
tU,, t,t-- 'l I v !"'.'
f _r (--\_n\,\\
Ronl< Theor Qrn,
. \ i jvj!Cl.I'r ri forls i\rel1t : * 6l it te r' i1r;g p1 g "= ll - rcrr.l* (A)
,* GqusS- Tordon Elrnrn(l tion
--> f e cj'u ce auS-rfler\l ed rr)o1! l)' e\len ( i''rr the'
A ffi OI rrx lg in redcice d rotN gchel orr for rrr t f
O t, ,S tn row Qchelcln form |
. tecdr^ 1
€J +n,r teqdtng enrrg ln eqcn non aefo rcw lS O 1
e eQch column conrqrnrng o le0drng 1
haS O,s everUNngre elfe.
( rt {tlslern l,f contrljl0l')* r Jolve loodtng vo'r0frlQJ
th term( of onu remolnrnq free voriobte{)
t-lom6 gQneous IqJtems :
no JcivliQn - In con€l!, len,
Lrneor tq I I sm cen t,)qv c
"?
Q unlq\rQ Jollrl tCI1 \--=._ 1s6.9rJ t€nF
-\
-> ,nf rn I'erg rr.loh!,| .tcrvrr<;n t \
r- FlO{t1C utnOCJl .ryJItr1r ci iriru,<it nOn, ..; Ott tct)j,Ot1 I lt.f ll,lj
=C:
-> qUgt|)l cc\ Yr'lr"irr r x
Li to,ri-r Lrllr ]
:r r'lorneogPneout 'tVf l€rn hclJ O unreve {ciuiicn
@ in lrneltg ir,o.,.1 6otuil(r\r
r[ [AI0] tJ homogenou{.f\JJt€rn o{ m tineoreqnJ
\.! irn n vqllqltr\[l
, ]!her s rn ( n i thoh the J\rJi ern
h or tn { in e ly mqnq \tci\JJ rart g
, Mqtrlx operqtlons
ij t A rS t'\.) x 11 r'r1 c-:!l rtx '. r"\'l -= r-<,!^-'s,
f) .= cCrl U i-\-l a
lxrn n.tq trix = rc: \-r.-r rrr'\c,r +rlx
nx I r^nGt l-t Y -- (:<li\-, t1 f\ l-V\ C] tl-iX
+a 6l es;cril)e ci 51:c.:cl f rc e'r-\ t r \'J tY\ c{ in'r'lt rt x ''
C\ri {> !el'i.
cl1t.Y-J .rl A rr-r CC '^-, c (,nci cLr\' '1,)rrJ I
J
C, rJ A is -5qL/ore i-\1c{trrr u{{th c{tl not\cltcigt:r.rol
er]trle5 '- (), j'- A <ir cr goncr\ r\a\ qt rrx
€) l:i A rs cllc) 5e,'-\cxl rY\ cllrr \( vvtl r\ crtl 't i..le eJlu<{ c-'n o I
€i^rlf t<r., t)!-: ir\ L] rf\e .. (rr\1 (j .'. A .= g(:<X I Crr fnCtifrX
tf /\ ig (rc (.r i Crr r{lci I i-r \ Wirh G(:cjrr cir ci-1 clr O
5(,i--.crt
I ,,_ A: rc1errtiry nncl+rix (i)
tf +v{o rY'\ q t.lcc-s3 <lre equcl t
--+ f rreS i-]crv e i l-re s orY\e s.i ae
oP' ea((egpenelrn g errtrteqi qr€ e g.--r cr I
ln <)it-\et- l"'/ s r<' 9t '
A = f <rz-1 lm'x.\ B = f ba;1cxs
{-er A:B <,:-> CI ft1 = i-. g1nd n=.s
qncl
@ cl
! - OrJ lcr crn 4-,)
ts $enerot egn oT lrne in tR, qx -i by
* Senerct eqn e$ llhe rn tR3 qr. t b\, t cz = (,1
Lrne0r eg n lnn vc,r io bl eg )( , dz r' , i(n
clrXt 'l Qr- Xz { .. ..t
0n>(n = -/r'
ie
cons I an I
--D .rqrr obie occc.,r on tV 1() isi power
.-b {€lriotrles Onl g fylLri{ I ptl *, tr! c (Jr)SI crr:l g
*t€ *o knotlr, *
tf 3a-Qg=-l -> SOIU+lOng CGFresPqn6sa<i Io p Otn{q ct1 ttne
Es,uJ ( sts)- rl(r) =t5-rL=-r)
( r, rl ( str)-t, (r) .= J-rj =-,)
et JC:
L .. t."
v's
sb - ug t
er - [ i
I f.r ir *5 s- T
3
r tl =[
b
!r?
r.{ q rl rtl =r js - nj
L
J
x thes e qre Sat e{ lcht In ,[ a\u'l P or
q m elll c f Ol f-c')
,tY1 e tv\,e sglq o(e equ,vqlen+
x,SgglaY'n eI ttr\eor eQns -- {tnlle {01' s I llneOF eq ng
( o tt eqn$ n ov e J tl1l P vor icl bie\ )
g 5gtul i on of o .f q .t tefYr o{ llneqr eq n's = Ve c i€r 'ii.rcj[ tt 'J lff)ui lQn(('iiq
.s()ilri tcn a[ €Ocrr eq.1 rn Jq I tern
ra Sotut (O n set -- se{ oc (tlt eqla J Ctt.. ttfJr,(, n-i {r.;d I {-ii-t
x 'r 66lvrng *he JVI lqv\^ 'r =,. lit!16g( crt l,DCl,r\ LJ J(rrLr itL,, ' JQt
, )F O.fqStgm o[ \negr eqns wr{h reot caefftcienr( hoJ etther
" unlque {olu{tong \ aO.,Ur 6 | ent ,l qJ tefn,f
, ^,. .t' ^, )
^olrtL
porot$o''' -' ln I rfls t QltJ morrl'lO\uil('nl
o oo ,lolur lcnt Incongtslerrl sq6lenn
$sivin g ( JtJji0rTl Oi LineOr eqn{
r ti^,e ilnSOr Jq$rei1r( Ore equivCten1 rf rh€rj hova if\E
JC{fflQ $CiuiiLn [0iS.
.$ e qul\,qleni Ji;16 = $cii.) hova lJr1lQue JCI! il$r)l
+ APProQcn io Jotving
. t[onS t c i r']'l g\! E.- Sq,tiem ir)t0 eOJtet C,t-\e {o Jtivt
a flrrn icr irrongrrCrr pcrlrein
( ,l pcond eq u o'l lor) h o\S ono lotrl val I Qble -rhan (tr t r \
sr'f rnol ono
)T qn\.| JOlurlon cj+ giYon JgJterh rd 0lJc '"|Qlu'lorl
)e On\.I .rolltiOn oF finotsg,fl$m lJ 0l'f o JGtuilcnof grven6ns
3" $rven39r,n- qnd 'flnAl 'fqJ'erYl =QQUrvarenF
,Methods $or Jotvrn Lrneor e nS
'+- iY-r ,lJ r r- , . -- .. .3 l^, .1 G rl: -:l-"^- i-'- l^{'
M-fr,\ {-rn f ';.rt g1_pcr,-i1 il irr r{-
(, ,lt'ra1 r'!r\L (J'J frl t
COi )lt rgtrlrq trrliti,l 5 <r
L,l l.)Crl Ciil
tQodrnq I O 1rr i-!l .tl tluo lvr..'cii^{ , irr.jr rf in l(,ai +n,,.J , r.,(t./,,r.i
en\riQ' ) !ltli.J
[oYm f rs ln a co [u rn r) Jo +h$ tef I cf orig leqdlncj
5161r tolt \ er\1TrO( \r tiOir\ tl
pqHrErn
L+,n .ltrc1 cct\-r i1,\r-1
c_(rft\<.llr-1rr1 q q tecr <d er:rf u q ji1
, flit cf r
enlrl e:r t) rlio^, I ecxcirnS <nri(j €iie (!
)a- Ll tc. tr)tll t(i!\, I ci\ oF el g-trQr)q
a, ttlel\jt i!t1': lr;1..r Cctr\ b€. pet fc)rrn€d c,.., $\Jf l pmr Of itn 651r e5Qh$
.lC ri'dr1,i lC,r1'\ r; .tO eq Utvol en+ ,f'rJ.f l efn.
r\lpog L] .il ii {l '.i u! .: 1'Lt ,! .!
li^r I r.:r'
\+' (!l\'rtt1rty t i'n l)V tl!')l€tc !Crl5r(il)l
6- A.rcl l\l!l,ll) 1g vr I !!\ r\- !t)a,l\er
--. 1.,(.(a'A\ ti a, tr e ':J rCv\ i(iriu (l tC.-\ \reou(\. r-t)ulir\ itj s,ct)etci1 t(n,'
r i4Olrlces A Ond R Ore rOw equ,uoienl' if rhere
qr0 SOqUQnCq Ct ete fnen,Orq fOw C,petof ton( iherl
COnvsrlS A lo g
L+ tl tngg Or o f oW equrucrle.nF -> c-a r\ lce hedr,..rce d
i O {Afne row ech\€lc}t\ t oYrn .
x- G Qr..lg st en E ii i1.}1fiti l I C,1
@ writt 0uS f r'1 rll i ril nt;ir,x O r* dr.r('G 1e r Q!^, eciaei orr -scr rrt
@ trso hocr {(Jb {r ilu +l orr rO JQlva
*-t> rqnK o[ rrroit,x * of non aero ro \A,B ID {C,! e! la crl Cf\ i Cif).)
ec. f qn( tA) : rqntt of mottr\x d . ,-., .hr.''
.-'\ n
tU,, t,t-- 'l I v !"'.'
f _r (--\_n\,\\
Ronl< Theor Qrn,
. \ i jvj!Cl.I'r ri forls i\rel1t : * 6l it te r' i1r;g p1 g "= ll - rcrr.l* (A)
,* GqusS- Tordon Elrnrn(l tion
--> f e cj'u ce auS-rfler\l ed rr)o1! l)' e\len ( i''rr the'
A ffi OI rrx lg in redcice d rotN gchel orr for rrr t f
O t, ,S tn row Qchelcln form |
. tecdr^ 1
€J +n,r teqdtng enrrg ln eqcn non aefo rcw lS O 1
e eQch column conrqrnrng o le0drng 1
haS O,s everUNngre elfe.
( rt {tlslern l,f contrljl0l')* r Jolve loodtng vo'r0frlQJ
th term( of onu remolnrnq free voriobte{)
t-lom6 gQneous IqJtems :
no JcivliQn - In con€l!, len,
Lrneor tq I I sm cen t,)qv c
"?
Q unlq\rQ Jollrl tCI1 \--=._ 1s6.9rJ t€nF
-\
-> ,nf rn I'erg rr.loh!,| .tcrvrr<;n t \
r- FlO{t1C utnOCJl .ryJItr1r ci iriru,<it nOn, ..; Ott tct)j,Ot1 I lt.f ll,lj
=C:
-> qUgt|)l cc\ Yr'lr"irr r x
Li to,ri-r Lrllr ]
:r r'lorneogPneout 'tVf l€rn hclJ O unreve {ciuiicn
@ in lrneltg ir,o.,.1 6otuil(r\r
r[ [AI0] tJ homogenou{.f\JJt€rn o{ m tineoreqnJ
\.! irn n vqllqltr\[l
, ]!her s rn ( n i thoh the J\rJi ern
h or tn { in e ly mqnq \tci\JJ rart g
, Mqtrlx operqtlons
ij t A rS t'\.) x 11 r'r1 c-:!l rtx '. r"\'l -= r-<,!^-'s,
f) .= cCrl U i-\-l a
lxrn n.tq trix = rc: \-r.-r rrr'\c,r +rlx
nx I r^nGt l-t Y -- (:<li\-, t1 f\ l-V\ C] tl-iX
+a 6l es;cril)e ci 51:c.:cl f rc e'r-\ t r \'J tY\ c{ in'r'lt rt x ''
C\ri {> !el'i.
cl1t.Y-J .rl A rr-r CC '^-, c (,nci cLr\' '1,)rrJ I
J
C, rJ A is -5qL/ore i-\1c{trrr u{{th c{tl not\cltcigt:r.rol
er]trle5 '- (), j'- A <ir cr goncr\ r\a\ qt rrx
€) l:i A rs cllc) 5e,'-\cxl rY\ cllrr \( vvtl r\ crtl 't i..le eJlu<{ c-'n o I
€i^rlf t<r., t)!-: ir\ L] rf\e .. (rr\1 (j .'. A .= g(:<X I Crr fnCtifrX
tf /\ ig (rc (.r i Crr r{lci I i-r \ Wirh G(:cjrr cir ci-1 clr O
5(,i--.crt
I ,,_ A: rc1errtiry nncl+rix (i)
tf +v{o rY'\ q t.lcc-s3 <lre equcl t
--+ f rreS i-]crv e i l-re s orY\e s.i ae
oP' ea((egpenelrn g errtrteqi qr€ e g.--r cr I
ln <)it-\et- l"'/ s r<' 9t '
A = f <rz-1 lm'x.\ B = f ba;1cxs
{-er A:B <,:-> CI ft1 = i-. g1nd n=.s
qncl
@ cl
! - OrJ lcr crn 4-,)
