we lar cera of a tealacs Lktules oF operation ———— PEt teh as etal APE cnayley Jor Mu aeiea ake 5 to the faren of a rectangular are, A_baving tot nes) Ad A Column vertices} Lines feb then A & cai OF ran! matrce each of the mn! no. # led an = aentot OF tne matnty = Phere ar, aA ferent notation of ¢nctesing the elem - § Me of the motets CTC) I te Bus duare matric «% generally us el — ord meet + mately 25 olay called 0 matetc of orgentonen’ a ea) eh em | 24, ~ 3 4 ° =r a ane eg = 2X8! “mates or pas order of matriy 1g 4x9. = AzA(A set Coy] = fa a, Orn a Sy aa, =. Gn - Ss aan Ss es J men BP the mada oF the order mens 2 ArPlecatiton of Merterees i= In algebra the matrices and then larger et Be tbbira yin tn tne of Simultonwus equation and tynear 4: Cur Sor rvartin, — “US for eg: the $0 ; acne + 4,9 + yz = by = a Gye + G24 14s z= by Wy + Mend 4%? > beAx e Ty pis of matress Os eg: Wi) Square Matnies— 4 ematme en nikten thy no of tous S Calted Sauare main. Obbrrurse 16 Sard tobe gia Az fy a 4 1 4 fa 4% a | : [a] gy =F tO Rel etete tA opts houhg aniy one sau And tiny noob colemat pHi a mate OF eden “inn! Us catted ous motriy, eq [2 = -2 9] #3 Rts Gasinie, Co luma meteve: = matee hav, boa ae aed eter come ea} s Met matress 5 > coy. a « { hub moter of 4 8 30 3 6 6ub- metry pf A Os) 2S 5 win te {42} stomaton af O2f S22 forth Columns of a és — iy Di'egenal matric Let us suppor thas M2 (Ody S men 8S ¢ matrie 14 fo ee ft Ly tin genat_me ed eee —_— eo Ff em al ~ Wiguan rately in. inich ql) -nen= diag onal élemenis are. are cxited diagonal enaine f ; Thins iit a AS tN nn Ne (nto teint 1S Catted tne transpose ofa — ——Selumns and the Colunan, Alor at or AT atrir and ht o: too 2 = = a clearly the order of AUS myn, ‘ 2a : thn “ene order of +1008 pose fi Motrx Avs mamPrebertiis of transpose matey 3 “ DAF phtah Cyrecnstal Mes Pair ot stracahe 2eere B = — 440 bab, atb-o ; Crectongular hyperbola ) . Propertees of transpose matr's§ . $f Aond @ he the to motnices then the 4ronSpose ef A aod O13 A! god a’ respectively. ues A (Avej7 = a7 97 | &ay7 = atat Ghe +rans pase 04 the braciuet oF the mig trier paimen cFenitien the product of their trenshace later tn the reverse order Symmetric Matriei= A sauare mosrts A = lees ues 4 motery As COyI ay) - tft Als op Cay $s eguat to the matey moctery Seth. This for ne sy Ory tagei stalar Motriy:— A clragona) mete'y en ore equel o Stalar Say Ks called 2 7 : Az (aj ]ain 1's Scalar meters ; — a eee —— ore 2 ——— rey a nqerdeiy or Pelentivey eertrye — A) Scalar metres so bite ech i Ee Dengerod clement de untty (ees) 6 datted cant enmtede on erty. motre A= Gti]nxn (5 unit mate of Oi) = f_ 0 then ej is i ta teen egee fisnugt fe De teed ta ca saomlES di ea Vue quiangutan emery A squcn crotrss ih Lkseh alt the tlecathis. — + bins the prenedpet oliagonal wre ter 5 Calted upper triangeda: Lr) actrees ; = Az [Oj Tnvn 0S Ubper trtngulan mater Hf Gey = 0 for Oj a 2 gt Stes 25 an Upper trtangulan matrie = be catty tower trang eileen ences =o ty the dvagonol oe terg ts Caled Renfinendas a a= (aj, Jano €$ Lowera spi il zona 72 mortrvy om tthich 2! tied tr’ongular peter 1S lor p Or tormtee setung _Equel Hatryi- Dito me Aand B cre said fenenin ogee Ehiy have Some order and thelr cori pending @leménis 7 ne element _1'5 co Single Element Meters 4 motery having nly one tlen tS colt ; gl slement mines 2 Single element matey Shur [97 os @ Single et tes ingulor £ Non— Singular elatn'eesi- A sq. mosey ¢s £ non- Singular vf tal po <2 Ac [eg 4 o=3 4 | FS © Singular mate gince tal ao Es et ‘i Trédtagenal Motedy'- Matry having non- Zeta tndeits only inthe leading cléagonat_ Sub ~ déogonel and clitergonal + —— In other words real maine 2 fan] ets tridiagonal ae Saeed) a egy ale Sy ay 6 6 4 = pea cere $n a2 apg J z —2__ "$a Fae = a Stostes ie ea —~E —=aarcc_aeet ratieny ef att 4 > 2 maeBe 4 Peto tv-9 43 scutg0-92 +4 $-ovev-9g no SiH) Saint mew 46s — 162 tn 0 Bist So 4 6 Ice + 14 Jay oo. = 3 2 a3" ~ 49% 9-0 = as [ae “Lan — 49) = 0 | Adjatot of @ square. Madre poet A= OT bee cour bimatny of order Hn nd bet '6;.' be the as aippctn A then +h Ara S Jawa ta the. mogrtr® the. fost or, the ot Aus caltent tht adoint of A. and es denoted The adjoint of a matey A= | p $4 tathen adpa — — aupspee a fiseat 7, £ [=r Pd a ee eee Ain 2-3 r Aa, (It+ factor Petry og Ay = Cr92{Z4+6) = 2 ta = t-199. (6-3) = 23 Aig= C04. (as = © ~UB-2) = = . (980) = 4 Azg = nS. (adn) = -3 adj A = q_-t -4] Ralalttetsl an) = =4 34 | at Fees (ies zd 993 = (lel tay = <1 aé-< Remartes- Let A be the square mein Pordin ty) then A-tachal = Alt, = lade ada At = waa ¥_jald¢o Tay a)" = tat G82) = erta-tq4 Condrtion t= Golf det 4 be the non Singuteenvalue of adj A Lihen the foctor 2 Cageg t= PEG i = 8 4 Feng, adja £ a+ ‘a Lo -é ic (29430) = 2 Oye Cat Caste Me Hes Sig = 198 (OB) = 0 7 N22 at (me) = 7nd Ge Fig a (-194e (Og) 378 4 Oe CUE Lb)= 6 “ poe oe he ee ee ees a rae) ai ensue) Tan 4 La =9) @ ] | Sojutvon of lénean £quatbnt— : ld Homogeneous equaicont— Att Us Supy ae = fi £99. Ay xtbry $922 GxtbsY tz =o wu tbsy +Qtze ele[Ax=0] uation = insat 69 —Non-homagenenu eger. the system of): G74 4 +42 = 4, atic bY tq2 2h Seong ae me Ry a, L euuraie ae) ia| ale y Siewie oa = - 7 (ax= 87 at = Se -_ AhAx = Ale + Ly = a-'g [*= 478] rf lal +o Luhin the hes trite! Soin. | bigs 7Seire) ol bomatenene of) Ax =0 "Alto zero] steal goin (x22 729) $£[Aleo then the sgn. ha pena tnwat sain (mon -xere Ing Sy siren OF non— hemegentous Linear gn. Sf A¥= 8 susie of non-heme, eMeouA lemear ean ~ Cases. IF lato then eon. tax Krtagtee Sat. ( cows tend) AEC 1 Oe Gose8. SF IBl=0 (asal-e- 5 ean: $n consis tent) fet Corfintte ayn, ( dacontesind Qe eeay 49% -4 — & Bay eee Sut sy +97Gnd Solving 4Lhi's clelerminant %+ Dd : Beta tsde and ot ete | $ Rank of a murtric (Rank methact) = i 4! be any oxo patie cb bas & Square Sub metrics 1 of ort teria orgins the atte tnants of the Square Sts metriteet a tien coed moirons of 4+ = 4 A_matrivy A) és said to be rank of © or fa ele “hug gtlastene nenstsca miners ef orden te" : : Wi An the movin of order Hts are higher than rte) + | The rant of Aa SA) from the definition of the rank ef +h reeled fatamd =a] j= — Remark d. | i 4 22 le |__every element of the _ °. Sauna | if pines — motn'y 29s minor of the ¢ order 1. or wate ee I | 12] [3« fz [eee [321 | cee miners of a a merle 7 T+ — J fe J aya a a2 Wve order 2. etetand {3 4 o a a oe 4 js 136 +3 6 | 443 e pats OSes ae oaeai lone 406 ¢: are minors af order ] a Baa rite pense ator eatin HL C8 non caro iN A dt. | Shere entsts at leat one mined gn A Of grcler'M tal latch tncto Gi Bier, » A aS sdi= cap be pinvttenasy a=4% ind the tank (f) of the matrix A= d2e2ie aoe 247\ Uf 40-424 -26 o00213 [3 6 40 t Fas ta-tag Z 724240 =0 lalz=o | = L wGela 2] [é to] = -2 40 TAla oD A= |t >» 3 a T au lalrns aT il eo 8 e 36 fe | — — at SG Rata 7 ; a: fi. ee CAN ae 5 mettny = 2— paticle Echelon's fotm of a matrix? a the non Fett ot Att nny presides the tere ralel§— ~~ jhe Oe ot tera precy, fired = tera elements 24 = (S$ l64S than ihe no of Such rh tere Wn the Suecedeng —— — first Men Zere ¢ bements tt O rani tea tinity, then if oS ia Echelon's find the. rane, maine A= {| 2 3 Bik esas = LA82 e Saas aed otAl = OS 2-8) -35 6424 446 6419% = | = —t =2f $24 = 24-24 = 0 Ais eo-factor = a 4 2el a> B~% el) ze = ee a ae) — fo 2 Py ~14Pa -F 1 t 42f 4 24 LAI: 2 -10= “ Vectors x, ¥,, ¥) -- ¥n £6 Sold ta be - re exists FF MS ao See an ( Tblen bets Kenko a : Mot all cero Such that beyati + Kata tKa 8 1 f eT Pa gR IPAS Gicinpe: | oh got kee Nir Ceeeeeeee Remark i= Tf a Set of a victors fs lencarty pendent ther 7 lenst ane member of the Sot con be exprened oS a tion of the telaining veeto J frmark 0: the on ond n diinensmocl vectors * Ng a Xe a _| ore tee early dependent if the rank 0-£ the matret = {. pong ]_tebith the gui. vectors oF toluma fs 2 Lege than M- ay + Step st astr +h et hercent toys Ler ees 35, flerminds of am fend he conk ote A = a 4f f_ A= no of Vectors Given the Set ofa Pe ed + of)_ ¥) 4% 4 Xp z $f X= [4,00] x, Co +,0] (i) 2% + Xp =¥a Kittie = Cyopo) + tao] +lees] ut hatte = Cope] +tetrel > + [itere, otra, osost] = tne = Steer aT Show that the vectors = [12,3] *2= - a Ine pemctent drt Seltr= the $l of co- eftretent of matrix se eS A i. fae Jeu — Ry > Ry +R —i7 i 2] — r3f — A= ifatae | eg ee, E esac i= UES a PING rerWvecger « — Hence the geen Sel of yestons U6 tontarly ena, at. @. Prove that the vectors ™ SiMe pepeel ees, a), y= (ea em ——frem—tentarly —dipindint éystim «Mia fay an, samunn tia thorn » 1 ee eaCur F402 + eo fo Si ° = ooo S o -9 -8 at & aos * OF, Pan? 649/ =/ eS ie 06 oe ee 44,20 007 = Fa, -@4y 0 to 7% apawertas ce a SS Tn me Shag ere) apestacs.. purge G2 o2k Sint =e ie Heme sale bf Hen cry ty +2 pire aie peaer anjotebrent OF crctny then ~__dhe —- gutnvetle esi Set he un saity depen hie Bam 6s +444 then $A) < re of vectors oe 3 ——— rn a ©ep ai Eee ta lS aap 2 #35. ion thet teeta = wel ee nd ee batt _ ares Fi ee Neel Pe Mie ee - ae =4 lt B. ere rant of A: = 2 froth the Bathe =lence, the qeun Set of the vectors = Mer, find the reletion bly them Ss Ss det 0/23, 43 he the Stalars Sut = [2-2 oJ fa, f —= 63 ¢) le. |= 12 a 200) | ws PS eta eA 84a +84, =o nis —______ freon urbe hee ee tet 492 from ean © % = -K Putting the vole of age and y= -k ah fqn th whe ay a, fe Hence, the relation of eqn. Hie ee ane SEX eee =o Skt Yat Wag eio : x. = Xo tay : Sho that the vectors = [1 paca= pn square martes A praise eee dype of nxt Sue 7] of 6 then Ren= cere matey pip the | | that Ae 240" Ve onsig ecegn veolons ob Go kenanccss seta yalue] Correspondmg to tegen value of d- — She: Chanactecrsires value [ Egen vale CS oleeng crete | pA be 0 are matriy then the proble to be the determine RUNG eieca la ar noe eee ect Inthitch 5 semultanecesly | hasilies the teuaiven: In aie Aas ao i AzaAz mT Axe AXE Axz= Ax — @) This equ aicen ty ¢s Knotin ah Charactentt'es oF pro blew [iar t Gin, +41g%e #2. 4 Oin¥n = 42% Pps ee ieee sag tS eo arx Gm, Hi + Om, Xa. 4 hing Hy +--+ Imn%Xn = ATH . Dhts 10. may be biriten a inadtror form ep stag ere rg geet ee Aro St nt ea CUS ta Gp ged ets aaa cee et ant 50 Gee Onan se mat Gained Moye Yan Hit o 7 oa ‘ T ' Grats 1 Ome * ane == tla -Adxn = @Dhe Characterts A-4Az Alg. fhe Chareteristecs value of A la-az] =o flutting the vate 4 the Mow “eqn:arth: __matri Now, AA! = Mutiep ung ®: Express the fellas a mated ou PE er erie ee anh ho Sie i ees 50 5 Sint cobybd thie! Let the matrix A mate cepond tts mate + charectermstics equativa ot 4 4 Ayia don are called phors eS ee Eee . my _choreretert Gt teS—— (A=az)x=0 then for tht wolue of 4 (a-azJxso 8 _gndittdeed ond hente the equalion— treveal fat? ed Charcroter's tec roots Lecgen value) of a 0 fend the etgen velues and corres Veeto Lt afaung marr: 4 liexee yal 1h A <2 2 Wo go71 phe chanacterdstres la-az] zo os: a yeafee[ se ate eaten a Al At6) +1 6dtirae a2 hI (446 HA4+ dz= = gait ap eg 4%, — 2fty = 0 —— Puticn itteng the vatus of His &- Lheoremt- Cay ley = Hamilter Every Square mates Sastsby " 4tts ono characteris tres 297 dey 4 = (ory ],,,,,—be 2 sauane motes a Oy Og == i then Jet Characters hes ArAL =Therefore +4 charcctecistees 9 fa-ar|_ to — palate. ayer shee Sines the ef tea ate ot he Can be LirvtHen a degree (nt) nae Thus, any (A=At) i? tetris palipna mal SNES) 6 ee - GON Oty 2 tej (a-a 2) = ~———Llbsre Be 8, — = Cnt _are square mateices o-f pectin ing There Clements be pel renal word = + — * babe lenois thas (a-az)- adj (a-az) = [A-At) rr” (afaie (Aqtz)- Bode! + 47-3 4 + On = (Poa% +P, art 4 (AgG— Gard + Apo Le) a obits ts on sdentity ty Salar de the Co etterint of (11 oe at = Pn zy . res ioe c el eee n-2 es ——Pre mutung ea Ale at [ man! + bn L Up by An te hove Anse, ean Cate thus the Cayley Hamilton 16 theorem Of this eg i