@

4.

Rdevatnk Seth.e^s herr\ re-x't :

Oefanvrrv nrul5 OF oRDER 2

Ttv d..e*t*mtntu'L o & 2r>

dunp'nts

EXAM pLEs

( , ) A = ) h t =

(2) B = fB l =

DETER,M'NANrS &'N_vE86E MAI{ l , t+E9C HA PTE R, '16

t 6u6 t t6-6l b " l ) l b , ' Z t

i

z &t, Qzz- CLtr&rt

( L )

(ir)( i i . l

(3) c .:

1g,, a"iv It Qzt Qzvl

\ r, "6\

Ict =

.gYgrenn or urrvnR gQunrroryE r,. rAE CgnnneRs RUIE

Cnneider +t,q ft1tanr,^3 St*^ c:onoish;^, "f tur?linear epfr^oJdfr.ns ,

'

a i , t1 t &n \zQz t a ' t + A "a

@

TIM, rn-cfrfiititt'v* , Tla,*rix of +^iA ry aystorn a(uygfu>ns ,.t^ SLtN/n W i

Tv\!, u,rhL,Q, $rfr

(5) To sotva {on xt , x4laca cr.tvm,fleo- efrffv;ur-nt rnaL*x wv&. t!r* veciorb ) *o rmtag a, ^erD matalx A r

A t : ( ! '\ b ,

(4) ff..',r/- t!^L delrrrni,no/.^t o A, :

b t & ,v l = b ,a r - -b , .&2 ,b L e.7y) ^

! o tr^-oP @ns{su&g

6t

(8) h?pQ

lA r [ =

Trv" {or. molu

X ' 1 = l

lazl =

hr Crorme:'s wJl- rA :

l3: ?:,1 =Lr@zz - b,ew(A l . t &r t Q 22- Qtzezt

ulurnn z sf fl^X^b vec.tor ofmafr't>. AL

o p A 2 i

0.u4.2-1

,ttle i5 :

t u , , b r - & r rb tO r r Q z L - 6 q 2 Q > l

l',16r

l * , ,I Qzt%2

( ^ l

,sls+offi oP equah,'o^s o%3

Ltq3 2

@

=r -36 = L lq

= 4s = sq

2ol =

&o

q q 53Z t-l

1Il5

l " v lqq3z 6r(o):' 0,+q(,

(e;ctlT- G)u;)

2 . DETSRM INANT,S d ORDER 3

Conoldet Kg mu|.r,ix 1+t\e

WAn

futto,*g 3

A =

To fnd-+l,w *opu

detep-'na-v& oP ahtvw no&,,;x , {ofte",beJow ,'

t fd(!, il^. k\At derr",e,rrL of fu6t fDw &y1 arrrd,*u A't r@u) aM^n J,n,(g16 . Tl\!.,n mul+*pe ey W A^_of W Wg' dlerrr.,entsi

Qqtr*')-@@il

3.

"l

T}J,a V,o ,*ug^d de'mer,,* of M &v eiz, ffilumn ),n

'fdIL W ttrild derner^P r

ddete +tw ruD a"d rclurn

beJow tI

lA l = tu t t l a * an l r &ng- ) !a , , a r r l1a6 la , ,e , ,I 43, aetl I as Qgz I I au ezz

= Qy (azzQzs- QzsCtt)- &o(Aoqzs-ez3Az)&n (ay Qg, - Ag16oz)

ETnMPLE

deterrUo"rJtt, of +'re {otle :''.;2 f}x)Fdm{t Vrq,rnahix :

A =

e

:

s ( ,2-7) . - s ( ts4q)a-G\ * 3( - r : ) +

5. M,NoRg w cocaCraRS t

Tho eleonut^ts of a- rnaJb&K *t rnar>ntrne aFfe*+lt b ddejn"an ru ceAt oA dewLbtd o&u.,,e,(prn a aubit*e:,n,ndn^t of & ma-ttix eoJh^d11"!' fr.!I:l0l.'. Thus u rnunor I M,defer^va,,* of tl^b Svbn&tti o, ++W, i+t* rou a."r& jt0- qtornn + A^E.g r let us @r&vu ,riW- ; @xay ,''a$.ix

I

It 4 u = I ? r , o u l ) M , r = 1 a , ,l Q g r - e B I

' l q t ,

A cO(qC*Or I C+il 't, tu rnLoOr ,tt,.it" U,s prusc:ik{6qn : Tkv ru(o" Sr &.L 6qn o( o'6ocJor

'is a/)

?oftou,,s : .v ,aj = (,)'rJ t u,1il

J o l 1 u I = e r ) ' * ' / V , , 1 = l M , , l

lC , , I = F , ) ' * ' lM, r i = lM, , l

I C , t l = e , ) ' * t l M , g l = l M , : lI bFACTOR L.. ADtrOIru vnr*,rc;

rnqhix is a- najulLxis rcpbteA lrf L+6

,!n WhLci- Wprfgcnbdorlc; i l . -

C =

Mali.ix Ad,1A-fho9

ihnnsposL s(

rnallLx. ,

I

F,xd"m0hir

r\-/ =

A e

' -z- .3

5

frl{owi,og rnat*.x A .-Gr'Ven t1lg

A^d fl,rp a{e."^tC

, l- l4 l

53

3I

q5

25

lg

I-{

fl; lLI

I

. - b3o

(

1 \

- _ - , L

Tt'v, fid:v)"+o ( c

v Ati A is il,* hn^spoto

A ) r * ^ lr r f l f i = v

5 . T!+E tNv A6E oF A MATRIx

f:pp61 +u,o*- d is a re,.L- nctrnbLr di,jtd,ru,*fl/s'^., o " Th,prt lb .r.l\L \s & u*IW" 11on 6d x-tCocLu 41^o* a.d,,-'rnvurce, O( c{ _ As u,x, k/rr,ci1u @,_a*, t v, rnafiix

r t^ W .LqfrFf nu_b*x I ',s

tt*- '

,ui+dpr"A o+ Lu im A"- r

(zl -For wha* non'ber ' Q,/ -ie

rerse, o! *+ oIf,o"- if .'= ( l o \

\ o t /i -

' - , . ' ) ' \ .

' i

+ ( S u t o ' o - o \ - = ( , o \ , .\ 6 u - z o r t I \ o t / '

/ ; \ ' ' / \

) ( Q a o ) s ( t o \\ 6 a - z t J \ o t / .

+ 3o' = I + O. = t/2. r,!or 6a-z = o

- ' .

T,O der"NC flra 's6yqtgC- o+ & ma&,iX , @cr&., 4ua , et(ourahq &rm.ulO- .-1 ' u r

A'I'. = t , ad'1A: '

(at _,Tt\L tubow -t4pltu t!^o*. {.,4, i.rl\luw A elisEa l ' ' , [ A I + o l , ' : . , - , ' ' ,

' " , ' .

t',^rlOJ','eL OP fr". frltArr; g

To #"d fl^- inuorse t uJe, {ti,Lt nead 10 +i^dV^!, Qe)efun\ttatt*. f4 I

( A l = z Q r - " ) - b ( t 6 - D t q ( s - 3 ) = z o - . t s + z o: _ 6

tz; I -ti ;t ll :l-E;l /t;l -{ r:lr . l , r f r t t r v l

L , = I ' h

l 3 q l t tI a r l 1 ; ' ,1 f??[. - 15

Ltrq

A4(A)

- J * q

3 J ,t l ts q

. _ul5tt5

5* l^ b

tAt

- L {{-l- l

t5

5

lgl r

64t,

Oa/,^ Oho,r.r- t

A A-I

To &t^e U*.fr>,d. UA u{artor

ft,An"SPoSA

tNa

I

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