Unit

ALD DETEPMINAN TC MATRIcES

Deqinitions and Fxam pler

. Matix

Matix aM arrangem y eumuth

numbex) in oue and olumns

Th mber are enclered b ba panuatese 6 broleah or daub ba

Exampe O | (3 4

23 So -2 2 3

D

Maxrx ener al rm

A

Omn Or Om m

A Ci) myn. And tMh. mn j- elemet ot ith row and ta coumn.

Y

eler mea

roler Mct Inditdea nuMbe1

u and blumnu y tus Motrtx

A: Corj)m

Aon A m byn mh tAan

Exoruple

C5 A 10 1

oCA) x3

3 Type Maree

Sauore Mah Nhntu

mber trluMns Mat ane ehal t Maix tallsd a uare Mahix

nUMbe wA

lumn number

TAAHhe matn a callsul a s4or M dor h h-<suare Mothx,

exonpe

A 3 7

t

A u a 5uare Mat Pdu2, A . Ro Mahi

Mer twve oly Collad a rw mattx or ttw vetts

ERaAple

A Lt 25, o oCA) = x 2

3. Coumn Matrix

tbne only ou Cotuumn un a

Moctna, Colud a uolumn maty or a

wtuwn vector

exole

oCA)= ax1 . 2erd or Nu H. Marx

all all the lmurs a Maix ae

2eo u Callus a 2.30 or onul max or a

ud e dnetud by

Exape O/0 )

rolar dy3

. Baua Mattee

wo Mahice ACan) and : (b1) ae TwD egua Cie A 8) ad nly y

y heve saue oley

d h) tu demetr at t toepO pOrding

oce ave

Pxomple and

7

A B

A B ) 7: (o, y>14.

6B.suivalnd Matrtce

wo Anatrice A a t 40ne Two rdo ad obe eguivalud y

ttm an e obtalns rom t o hy

elnetoy ronformaton BraMple

DE u oiter as A B Ond od A gua lud

A ) B 7 2 . 8

6ld

A Ond B ave u va at

Stnce

7. DtagAa MaY A 19uae mahe al whre elsmet

encapt those u ta prvinupal uodin 6 Mow diagn 2eo calld a

dagonal Mocht Exanple

A 5 oo o66

B are dtoa onod mahices

&Scalaa Mahi A dtaqoral Muhà wwrh al Hu

elamas tu dl ngsnal Unaeud u calls a

tolas Mattx.

plee. A

tt

.

oo a o O

wure au onstaut.

Vni Mat e Ddut Mattx,. A 9uae moctt wlre draqonal A

udh ArtCuntty) eat aud nm diagt

ewwwh ave Calls a ni ma

and a dandal bi u an tdudity Mm

alphae I.

7)

example

. J unlt mar bader

Mah y6rdoy

3 (a,) a unMocha t all

l, SymMdte Mon

A 2uare Mochtn2urh had

O i all and J,

tolld a 2y MMAthe Mahhn

Bx Dupe. A 5 7

7 D

9 . Symmeic Motrà

BRUre Mat 2udh te

Oj j ter allii

CouUdakew syMMihic Moct da 7

Ttanqukr Mart ,

Toongular Matice t two tl Ppe Aoaqula mohin ond oue

tagula Maxhnn A Men A (A men PP

a thvtong ula Mathy yte . A MAtrt A (an)rwn a t wer

ag ules Mata y A Bxample

4

a w ilas

N ain

13. Swb-mahan ,

A Matin blaind by doelatng su

OUR 07 more tlumn Cor bo% MUre

aubmatix a g Matà.

Eyample lo S

D

Snbmatte A ave

4 3 4

D

3 3

(1) (23t (o) and 40 on

(4)

4. tdg0na! Matdx

A 4uane mah A&ard be a A

bhe gond mati A'A AA = 2

whuve A e rans p be A

BLampe A =

AA (Y a e -2 a 2

2

)Non-nqulas Mat

A Guac Matin A u said to e

ADA-AAgulun TAl#0,

whre Al dnste t dudeyminady Ni

A44uare Mddn A 40ud lo be 2inq ulas

yAl 0.

Mat-x Operaons-7

Add Hron

A-Catmxn nd BCbr n

AtB- (Otb),

Eranple B A

-

AtB 2tost/

t2 Stc

ubhraion 2

B (6jj)mxn A ( tr

A-B CaT mn

XohpA :

3- A-B I-1 4-2 S-)6-3

:fe

3Sealas Multh piceti en

Oud k oa 2olas A CAtj mv n leA le ACka )mx

EXopl 3

) a (3

12 (s)

t Muth pcothon

A-O mxb And -C b,pxn

Than predet AR tu Malk

ctmxp deptrsd os

ohoneC ApbtGiabattbfer all2

BxaMpu 3

AB xI t x3 +3x 5

xt5x(-4)t 6xG 4ctt Cx+ 6xs

(13 -8t 16

-2.0t 3 tt ic30 12

th) A-Ct 23

ABC 3/

x t x t-s)t3Y

-1ot (8)

(12) BC4

D A

x -s (x4

x x axl-s 3x t-s) 3x6 3x4

-to 12 IS (

BA 4 /

4xt (-D*1 t6K3

-lo t l8) ( )

AB BA

5. ospS

A COat)mxn en A C )Rxm -en

EXAMp

A 2

A

Atote (

23

(15)

Proo pehes

Addrhon D Mahtee ) Commiekeh ve

A and B Ore

rdaktUn AtB- BtA.

&ohs

Henc Moclx nddlhon Com m ua h re ence

2) Assochati ve A,aud Ae moehe

CAtB)t A C BrC)

Sespec oCalor oth

kCAt B) A +E

Exsleno Ddandig. hut hult mal 0 tu vder A

A

ttu c oddihe datty t

d 2atsfes A AtO Ot A A

(v) En)leno Dnvene

Aà tt Dddih ve tnvense A

A+ C-AJO - (~A)t A.

C18)

(vi) Concelorhon Jaw

A,R ond C Orne Matrice y tu

A+C tC

Mplteg A= B

)Mutpltetton Matc ) Ao Comm mutative

ahrtc A Ond P AR nd BA Anhght heve bean eginab biet Ta Aaud hof be egina

.e A A OnAath nve e

C Acsou ce

ARC àdlainad

AB C(AR)C ACB)

D Mu tti p tahon ddLwhe w reipet

addhon

MOhhces A A And mXh, nxp and Axp es peef vely

One

ACBtc) ARt AC. (rv) Ex sfera Dosnh,

A a suae ahre Orolas,non T rolatity Masndx ans

ATn APnA

C7

VEa)g tena vese

A a sUae AMoh dor n

enùb a 29ae Mahi

A

oe Ond hoh AGnar,

that odor h &Unch

AB

A vese

tverase Cnailasly No-o

.Thae eap ho dhveee s)ng ular

Mot

BAT and A

AB O (null mahm) does not Iply

A 0 o B O

Oodor mxn (vil Mocltn Oodor mxn lves

AO Onxm and OA= Onv

(Vi) AR AC olacs no ply B C

Cle

(i(A')A

CY) (A-+B)

Cwune ALalas ( (E A A

A u) (ABY A

(AR C ce'A

(3) yete Crnaas Egnohong

Co sdLur te utors tb, ys C,

Ca e Cous wide

b

ba X

C

C ,thy AX Tau ,1/) en ngn- ba y Dy

Hen AX-C. -h

enaiOns OPnAi losly tu ystem

4, + by t G2 4 atb3 y t% Z d

ds ty +0,z d wTen th sf Malt rm be on

Ax C

(.

Dohone

C A /a,

b

C3/

) C

o be ) Aystem naa

CoAshstent yt hay atlaat o ha &tluton

eguato 20rd to be

be UnLAs euahons sad to be )A

rluthon ho ho thconss fent

wignsolahon yA/ o, 1) Asystem hou a

hoh-Stng u las -t

neas epustiohs u D When C0, a yster Collok a yserm t naas hBMB9eraBUA

naar [v whs Ct Ax:C a calda qstarm naa

Abn-oMB9na bus eguattohe.

eg Uatir

C)

(vANhen C-o, a yepm Tusosegtuahos Coulld ylern

Rer 7 oy to eguchesa, Af:c (vi) Whur C fo, a

Detesm) nants

ost errdo lerrrn paud

The voune

sIhs olnn tt s ely.

5 S Od-bc

ba -/b ba3 d

ba3 ba3 baa ba3 3

xoMpe

6 le- 4 4 6 3

C-2)/3 313

3(p-C-)+ 2C-0-)+ -3)

34) ta(3) t (-t 12+6-/ 7

(22) Prro petTe Delers rolnat

Ci Al-lA'

any two adanl rbeu g ctre t edborp4

brt tto n ahanges S)nntoa ausult hedds fer bt

he vahe ey tae determ) nad enmoAE TO 2asn

Cetm 3 -145 6 A2 2R htercharg d

7 7S

Fr) any teco bwg daterrhm Irat are

Snat lo eg ult e Cxoumnk,

oluhCal TO Vahnes ty t drerminaet à 2e0

2 23

mu lyplGed T elohmens

e mu mu IApled 6y k held fo urntns oe by a Unnatouel k, tu volus yTte orlesm)nant

by

Oa Os3 Ag 33 A3

)D to TA elamLnf a vbo( OY Column) b

a oterg m Inant oe adhed E me tea elmed

y Onetter B- 6T Cotnmn) *he valu te

ohouge dbes nof oclersmlnad

Ata, t ba3, 1tb 3

2 3 a33

a celen7 oa sherghrm 'nant

Sum yeoo elenmsr tee datesm )noaf

delernnnant

(V) eah elsnmeut a 60

eeyprese Ooun be

t b f b ay 3

C33 O3 2 a33

Counn (V) evey Voahne Arern)hal y 200, Vane 2e80, Te

a12

Coamer nle br Doler mlnant Method

Cohahdar tte AysYem neas eghatens

Capree Mota

Ax-C CoOrm mer's hle w

o-Lut io n

(A Ay 2 A TA

aroe A & obrlanad by neplacunp ha Coeft dent oeroe

1AL AL

Ch uumn A y the tonsans (C) tne Letur

yete

Lotuun

A obtan4 slmlas ly Ond

(3) Prredetd Deterrninanf

IAl and te1 ave 40n d Ha Al1R) ebtanal bg muldipyy euty st

t d dobrmnend and tne y4etud

mu lthpled pars aud a olctexmlnort ane

007 ten appogoale pleda, um

th dledexmro à mu hpbe

by st yrt bo t e tohd dctevminentt and

ploced sft (jHe pechon tu podu c

aud

predecd

oeterrm) naA

2x3 +3 2 |24213K4

ta6-+3 2- 3tY

f16 9 8

venag coti

38-?=S

92-12 o

sC-10)e -ca

G 16 114x7-18 X1 =]I2 - 82--5o 18 7

16

CAMd

1

bta tined y delo fing

twhio ContaC

diuced b n) n

eltupd ar; U diueted CopaLtor

A A C-t)M

eue A 7j:2, +, 6

Nole

A -Mij Tt] a odd A Mij

xaMple s al

minrS ud Copac Find

se u

Scluthn Coa Mino

t

lolO

Cty 2)

(3,) 3 3

(26) Xape

a Cnd toyut ferns mno Fnd

7

ouhen

Min Cofate Eenout C.j |! - 3/=9--0) Cte

6 OmG46) Ct)

Ct3) 1

54-28 C,1)

7 44) S

59 Ca

-33 C, 3 |C Ro- C-12) 3 3

7 8-7 2

(3,3) 718 (S

(3

5 Cnya efer T 6*factE 4

t7x Cofa tor

erm )naut uair/ 7

5XaI+6 x 7x

56

U. te tond

D xteftfb tfer ) trqurby

ga ter y (-3)t A ojactor

(S

Matx Coa eter

LofA Lter Modnee a 9ae ma

by Cod A. A osubted b by rploxeiry E Db7atusd rom

Cofa Coa o evers y Ltn

Eyop A s 7

R -32

Motri x Adyont

odj ohvel mot

matr A duotd y d A

byfA t cbrla Inod

btlauwih tep, ( ride teu rea poie 4 e

Find tu b ya ter matd A

Examp la

7

A

Ca fer A 6

t

5

6 en adj A ta A

Prrb pen yeg adjA

A.odj A) = /41? (adj A).A ad (h B)adj ). (adj A)

2

Exa Mpla

ad 1 26 s

s9

- 2 A ad A

33 S

ts5

Al2

Hatet OpevefeAs

DveeyAMir

TA he cund mly y A n ahgulos e tAlto

lanetd y A mly

N tspe t ejuuhe

A. AA

Au (A)

A and

5olo And andR R enufy

Method thndiey hvete y o Mahy

A M) A) A

Metoed

xope Matl Fhad t veyse

7

QorluhenA

(3/)

01 Ceyectovy Copocfe0 Mthon (AC- ;)

C,j) ElenoAt

2 t- 5-7

Cr,

6 3 Ce2)

tt

21 C2, D

4 - 7 4 C2,8 -7

Ca -6

6 8 C2,3

-18 (31) 6

-6 5

(, -7

Boyed bn tins o

6 AT od -7-

6 18

A A

hd tu hveye

Colutton

A

C) AT' lal:

Ct2 c

IclC b

lal a (2,2) d

d b od A

b Ad- be

dl A o

adi A

1A

d ad-b

EtRd GVerae

A tD-6

-o:SO

D-50O7S

Meth ed Pindy A om o e3enthon sotteped k

BxaMpCe fiod (o (o A-50 2 02 A s

C-lovn o A -50? o

Past multhpty ty9 A

to AATso 2 A7- oA

to 2 soA 6

SoA: (o2 5

7

O.2

6

-2

o 0-2

Examp

A satiste steu e bolo n s,7

Ae-sDe, ound D oanetu Tu200 Moshi iod tt thveo 7

6. -0-6 0 6 6 0 6 o.

Exewpe 4otfey

esuthon -6A a4- y7 5o. Henu dludus

valus

Selhe A

Arltl +/+

++1-2- -- 1+l+ +(+3

6

--5

6 -1 2

SS --S &Stco

6-(o 10- S St S

(0+St6 -S-(0-6 S++2

-2t

/ -al

CopgtclssACA94- 4T

1 3 -

-36tla- yRI+ 3o-9 -301 1

2-26IG4-e30 21+30

O N-re AA -Y2oNen bof atola

Pot mu tdipligiy 5A TAAT9A.A 2AT an

A 6A+92-AT o

S4A-A64- 12

AT 6A4 1

6-2f-5+470 -6+0

-5+64D 612-9 -s44

5-t S+Lt

Cbae ATA-D2

Selvngasytem y mn lHenised oas

egatrona by Dhvmye Maty rrthy

Madty Metthad

ven yclen guotons

C Ax C

wbhoe A-botRUt Morlhe

Unkhs tumn ve efer

C-ConeHavs Catumn vefr

X AC

ExaMpl Cetve tu epUotho by Mah Methad

31 2y 14

3 ot+39 8

Selushen Lthen eguahu t Mochhe fem

AX:

A ()C)

te n

A

ad

:9-63 AL

(7)

adA IAL A

AC r4

/3r 4-2x t4 3xe 3x1&

3-36 425

(

Vextgsti

3t y:3 (a)+ 3 ()

33y 3(ah:(+) : 6712 (e

Henu 23y-y ane Cerecy

( Mabehe vei Madbad sefoe

elltn oydes yefie

Jy+ 9z yt z9

Colutron Ghven quation Co be tro t

AX C

) ") e tind AT

( A

oty Cr) Coyotor

C,j)Ay A

Ct,

Cva) Ct,3)

D ta,)

t,)

(917 A)

C3,2)

CbyateY

A ad

tAl a)+ l(-2)-atC-t) Tvsv prodoeu) -2-4-2

A adj A IAl

AC /

/2 ()-2 (2)-( o (t)-1a) t1C4

Serlndion n y-26

eMdamy Operratens OvE(uwantasy ovsyarehn femtamy Mohh

Eoth tu tallbuw tng eperrotloLs u callsd on

eunkany foronnsters mathon

Ch Dterehong y auy wo 6ws

(orr 2Aterrahona ony tuuo tum re) mulhplhon

7 abw by a hon2eRO3Colar Cmuthp bLath on al u elsuen

Clumn by a non2esO calar) trs the oddihon totu

scolos mulhple t Contres po rdirg O

elonnsnd Anetur on H 3 AM urlds ubracitoal

D tu oddllhen te tu

elnh nstor rtunmn) Czluumn a Colas mulHipleste Ceresprmdiny

Bxaple

A 7 A

414 19/ RRt&R, 942C7) 92/) 5t2/0

14

(+1 Rank

Deynon

mohe moxhet Tt crds

Yu Jas g est subn ahr?x twbre daterrm)nan

Rounb ya m ataiy dnrtd 6y P(A)

Hedhod

Fnd tu oan

3 6

Sluthen y-2r)- 2( ta- 5)+3 (10 -A)

-t)-2(-3H3 (-3) 6

6-6

- 0

Rane ey tt Mahl

roAhle

9etnthen 3(8-0)(7-0)+ -)Cq2-32)

8 3/6)-2(7)-Cto)

a4-14-10

O

Rauk 3, CoAs Yols ubmachrin

(nRonl Null Motnri x O

ROnk C) m, Ronne y n maht x ordu "n

m Connet tCLd m

Fnd aue S 4

7

-eq uare 3ubmahrdes mday t

Feu ornday3 t

-a3 3

3 41(7-8)-3 (7-)+ 4-3 C-D-3 t)+0)

34 7

Mesdet a7- 83Co-2)t 1Co-

2-)-3-) 9-1) - 3

2-t6-

RDnle

(48

Methad7(Ranlk 4oom an equ valuut mohrte

EXAMpl

ronb 7 Ehnd

3

-

Soluthen R

-I /R>R-R-

o Rg3A-R

O3 7

-

Rey-3Rt 3 3

-1

P2R R- Ry-R-Ry

3

Sin 7htd eo 2 busth Oe ne 2e10t Sinu Hhone eniurs ho )rd D yath d 3ola

Abbsingulas Aatrix

-3 0 Cbnsdu

Rank tf qluen adix

ey

Co raY enty asys em , Pmalanab

Ax: e

Ranb(a) RonkA:) RanRCA): Rank(A :)

testseut Syeen Dhcohs ofeatt sytlem

RAnb(A) 4h Rank (A)n (ha

tol

hn eldhon nnde humbe

Srluhon

A syalemy imu Honsstd laa nohens

CDsheee Y OUd Auguaded Mahy ave al

() ank tyA Aronk y (4c)= nCns,tg Mnbntet

f syele hoe a unige Coludhon

00 Aeyounk7 (A:0n auk

hos An gne nuwbe

t Aratiba Tnk y A oonk g (A, e) u sylem u lntonsgl

e)

Test Jos Constden ty ond blve, y Colabbut Tes ExAMp

A 3y + 22 7 y-4z t 7y a

Svuh on ayven )) ven nattohu Louu be tura

Mathi y

AXC C

A o here

-7

CoAs ol

R2 CA C 2

-7 RR Rg-R- R

3

31

3

D -S R3/A-R

30g-0 O

1-5

(e6)

33 -67|,-s1-a R1ok

S

R- R-3

ShA

Rong

CA,) (no. ty uner H

ba yeha gnavtis&dteleuc

-unh 0n

CAnd

-13 Seluthon

y

2