mathematical methods of economics notes cum cheatsheet, Cheat Sheet of Mathematical Methods for Numerical Analysis and Optimization

Download mathematical methods of economics notes cum cheatsheet and more Cheat Sheet Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! “fh 15 Determinants - + Properties alt + It we Interchange oe at wal vemainds wrapfecled ‘ » | | Lumns) of a det, ts 2. Hy we Interchange ony Zvows (or co ) of 7 cds A {* gets c ar . she (or colle mr} 42 0,a detoaminaut var ° eal or propesh'onal ’ Hur rows ant columns of) 4 det , a pe a, iz 0 tae vf A on | 5 we can Talee ny. nom 3240 muunber common rom a ror ( | column ) ef, a af, | 6. Faom ond now (Orv coluuvur) , we Cal any otier vot (or column ) subtract o add k Himes mM z “a pT ¥ oa non = {2 4 s/ti24 3 O° 4 3 o 4 4 o 449 g. [abl = [alle ‘ + produck ef 10. Ye det. off afagoued [mahi > upper Ar o Lowes dv] 18 Bragoral Blame = b Positive vouttants Ques tet D w a 4 » wher a aud are p 200 4 d b ab amas, with a> b what ebvaleat Liner tr the x4 lone is d= 0% i beat peste 4 plow tz D>O > d= -x (nab - ga) ty (te-g ) = -abx> +4 + - 4? = -ax | bx - ¥] +4 [bx-y] [ bx-4] Ly-ax] o » Liter (bx-y) = © om Cy-ax)=0 bu = ¥ qr ae l = og " Scanned by CamScanner Henee p= 0 along. He sbanlets Lous free aud | ne 2. ") bd>o Cbx “pg y-ax) > 0 CP Hrer OL (bx -4)>0 autrol Uy -ax) >0 Cox-y co fy-axjeo 4< be yr ae t > ¢ ¢< o* % ae <ycbe bus 4car® but arb ts given aA arb By xP7O Pemee au mui be nigatine 40 D> when I) xKO 4 ax <4<bu 2) x>o F bx<ycan . Ques (13-4 /10) Prowe Hrat bn= Jarb -.. a a AH .- * = CL a arh b” *(natb) , » Dn - att Qa ....8 a atbh..a a..a- -. ar C, > Cc) +C2z4---+Cn Dn = [| natb % --- a natb Ab --: a ‘ ‘ . ' naib a arb = natb 1 4 A 1 aH + O 1 a te Ri RE- Ry where 4 = 2;3,.---7 Scanned by CamScanner 5  UtS Arce te with cides farallel tr Hig anes bas 7203 ak (3, -2) aud (-4,-1) » vertices Fino Hw ora of tee meetaugle |0,0) ay t———+— orca eh the - 2 x dreaek A ABC -1+—— (3,-2 8] , ) = oe ’ 2 | 2 » -4orT | 3 7-1 bt chon 1 C4-F) ([28 425] -![-21 +6] 41 [21-8] : 449 #15-29 = 35 dq. unite +E Mapai Inversion aud rts properties -l od? A * a od yA a Properti e+ 1. Matate A te Innerti ble ols it it fp, now singular Le Unverse exist) only if lal Zo 2, (AI)TZA z. (at)'= (Al) 4. BaA)t= atet 5. AAtT= TATA 6 (eat = 1 wt, be comalanct sete. t. Invece af a. syria, mabrine Es alse egenin “se ban a ee matal x Mp one fn whole Rune ts exactly, att auch an enth soln. ee one mon gure eukry in | me veabtble pusol describe fh Scanned by CamScanner wor > M= (3 p) whe arb po To shew M ce invatible ‘- IM] = ba-0 = ab Zo he moe ebgubar, madre heuer , invertible . New gud its Inverse . A tynare matin tr upper trlongular if alt caments balere Hoo parmcepad Ai an , Abow Hat a 3x8 bal matyrn witty no tems on He priuct pat Ltngetal ib Ierbiole and oat te tovews te also upper tai - Oy A Ae Auth. Hat aie fe » A ( 0 450 os | » 64 gnen geno Lave ve tet 0 oo %s prtefpat dfagoued IAL = Oy %2932 # © > A ip Invertibte > Find Invece ofA. Quer (13.6 8) Suppore A , Part D are rare matslces suk Hat A= Pop? . , 4 Q) how Ar = Pp> pt b) Stow by Induction Hit AM = PD™PT gor ang m>o. Scanned by CamScanner > a) a2 = Aw = PoP TPpPT - Ponpt = pprpt 4. b) Lek S(m) 5 A™ = pp™pt » (4) Luk = A= A Rug: pppt <7 LNB = RHA > SU) Lp hue . e Let S(k) bg dome » AK = ppp? * s(kH) tug 2 AKH Lyk og - pptp'tppp? = po* roe? = ppt4 pt. > SUeH) ts tue who Sik) tp fue. > Sim) sz tue VY meEN, Quet (13.6/10 ) a) Suppose C be a darmre matte of order n tat sotiafyes C+C= 2. thaw tat C bas au Tere aud CTHI4C. b) tho feat e272 -142C aud ot =2L-3c. Dalc*sc=1 (Given) Muthiplg bg ct > > c+1-c1 b) eh4ce 1 oven )-— 1) Mute pls by c c4ecse ° C4qi-c-c€ G beac uaing DY = 2c -L Mullipts C d eciacct= tet 3 cl4+1=¢ ci 2c-€ = 2(1-¢)-¢ ra 21 -3C Scanned by CamScanner Ri > Ri + 3Rz R> Ro ~3Rg ' 06 1-3 2 ; 1-3 2 (: 1 0 cA -3 Zz y A 1 = [eg a oo | 2 “1 0 2 ; 2) Rank of a Mataix The Rawle of tie matatn, Pe Aehined as order of tongest nom vanieht ing minor af, Hie matalo e Max hank of Amyy = Min (m yn) 0 (A) = ala) eo Ehemeut ALO fous do not Hw nauk. of a In aarnplile 41Uke Hoe marin number of; Lin ead Pndepen dont snow (or column) vectors fn He matrix ds rand eff wataire: e Rate = 2 44 Quis Find + x= [5355] —_——-_ mat rin dot moe Harn zero elamuls & kounk > O . man = 2%4 >» some (mar) tp 2d. op tk tas 4 in dipendent aps & Rawle 1h ; tke dus tab a Indepurdent columns oy Rawle tp 2 . » OR ¢ AS gu (14.2 | Example 14.4) Ques (i4-2)2]e) nebeuntne He Amik. O(a 430) 3 25 ih Scanned by CamScanner  —_— = n => - 2 1 3 7 , Let A = 4 4 3 ) 3 2 5 U1 3x4 Max aanrk = min (3,4) = 3 Rr > Ro +L Ra > Ra- BR 2 1 3 7 Oo 42 ao | o te Y, Yo R32 Rg - VqRe 2 ' 3 7 ° a. Vs, | 0 2x4 o oO °O ey man nade = 2 " Le ap] FO 2 tome Adee Quit For what values of ke will Hie Amuk of dooG ye [: 3 rs | bo au 3. 1 10 |T+ék & 743K mxn st 32X38 By man rank fe 3 + Raul be 3. be miner 6 rd fas th deleaminant Ve 1 3 642K vole non zero ' 3 10 1196K 7 0 i 4 JH3K 1 [ 10073) -4 Ut4en)] -3 | 3l7+3k)—4 (642K)] + 1 [ 3 ité6K) - (er2k)] 7 0 > 19430k-68 -24,% - 63 - 7k +12 +k 251+ 18k ~ 6€0-20k # 0 “3k+2 44K 250 B® kK 7-2 Scanned by CamScanner KUL oe vs for vank to be 2 [A]=0. anol det of, ino 282 7° [al 2o yk = -2 aud o% |' wol7 to“ apart, wi be 2 when k=i-2- _ Russ (Example 14-6 ) gus ( Example 13-20) ques (1306/13) gues (13-#/ 12) Res (13-2) 4 ) gus (3-4/4) Qu (13-1 [7) Scanned by CamScanner