Matrices and determinant, Cheat Sheet of Mathematics

These concepts are fundamental for solving systems of equations, transformations, and various applications in mathematics, engineering, and science. The notes cover definitions, properties, types, operations, and key formulas to help you understand and apply matrices and determinants effectively.

Typology: Cheat Sheet

2024/2025

Available from 06/14/2025

harshit-tayal-1
harshit-tayal-1 🇺🇸

1 document

1 / 14

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe

Partial preview of the text

Download Matrices and determinant and more Cheat Sheet Mathematics in PDF only on Docsity!

yey as & | a No. ow week ; = i X Nook ~ Nee = | Rafa jom 8 SNMEMTs T Rones” eau ) eleaneds = | a TH, Pwipal peg. 2 1=\ elements) = tr} (Mee retements J < * Speoa! ¥ Modi ces * t Sees BER Tan Oe Row “nor. ‘oR Row vector} On \ \ Type 3 sm 4h 4h An AN W) Colum nce A= on Oven C) ese or putt wok Nt © © a(2°° A= oO 3 ey N= \o o “|e © © ‘ NK VY GO o Bev 66 © Sr \d) Pork zowre\ Mod audes = mn “vr, VO7™ => ita. | \ irie.oF co 7 To-0F Row 2S 2 S Ce) Nexticoll Model » \' . = 3 6 me (No.of xows 7 nio-oF cal”) a Scanned with CamScanner Notes _ lee — ~ Ul) Mini warm queer of zeroes TH Oo MPRES OF Pane og aed six c& osdexs a= \* peer act Cn) = ro Sq) of eypvers nwa dregpual | d IES VETTES II TID TI IAS dA h ) W\iu\wiu wr wuwber AK eb oxdes 1 cotoos | uwit+ ano a # fal eee a Sys Madris te Slogan, Nonosicle De TON More x4 a Ke x usre-CoVy iim every saues ed oy replad “4 eim(n-)=% cael ass octake] a Pre metry eerxexm™ Want wht F element Covv es poudi aq Ww a, dekevm\nourh et Same. selene = Voz Detexwivoud Bo AS (4) => \nler werlh) \. \ | value of: 4 am te, \= \ *\a vag (durée A = ay do-dy ~~" dy Square ee ae x ENE 6G K=6 2 va ms °,| = Ads But \Onw \EO 3 BEC Garg ro +) SE A is 0 Sepwawe wud Be Ja\= o 2 4g woes Moder 2 ne aoa > Nowe Sago et Scanned with CamScanner / 4 SI // ee “ LEeEoooe bbe eee d dd dd dddII IIe’ IA) Trac6e oF a Square Mote Oy Or O13 : Diagonal Evtxtes =\ Oey Ore any = ? \o aum of “9 - a Char Oa2 O33 (oul, ~Yor Ca) = Antr> Geet Crag = Fay = &. As Coane n & For 0 Square wodet k Tend = oOo regan) = KOS * Te (age) STA THOR) oF \A™ gevides ih tute A @ a X o=(pean) = ox (cB a) X *& Multtplicedtow of Morrtices t (now by column) & “Tage qmakeices Ae E-B awe sald +6 Ce conformable For amu lipttesiion AS \e otayamem *& OCB)ENXP Le War of column WA = Nov of Rows un @ IE ota = ~ x), AB) = REP Men OCHE)E™MAP | a ag Mulltplicohtion ts dove Row oy caluwyy ew dy On Ow As Be Yor by 4, Cc = A\axs : Bar A -(o., Oor as ) =\ boy bop .— Oar Az2 ez 2G ‘osy bar / 3¥2 Cy SB Rows TD colwuvr Anion + Ore! rt Osh) Ay Aybso + Ai, dao, PB= Cape = |Aar bu *Orr%24 + Aas) yy bit On bre + Oratore rayon + Os2°o4 +0303) Aqbi2+Gg.bs24+ Ogab32 Cr = Ga tayy + Ogata Aes bso Car arb, F Agabeo HAgabsp © Cae Oy oye +O ba 4 chsh a Scanned with CamScanner 7 yh clus 2 Also (ax) iy - {i ewor A of 8 “ i c product ° a VE we wont (ree Ceamptere roe) : tes _ AB yen - : aa) of aa Pugh wore | vee : ju sl ¢ re wk € ye cotwan Ccomnple®® col ») oP “I # WF we wort ‘ a] AAD ymem rs eolwwn = i Rul wae | « Com i ; \' of # of -@ J a eas L\ yok ) -~\l—_- a 3 \3 at - - ‘Pos ar GG, | ¥ V\ 2 A> so rn=(Ret \sh) (ee ) = (© ») (! ) <>) i ee + fet (OES) = G9 (5) > Gy KES) : = V1 FV a (5 -333 & Rew A ot Qroduck = (2 ne A 3 he ( », ct 2 Row calvoan ot prod rv \ Ve (3) QE Gn 3K % Proporres of Mabey Mulbelicabtoy € - ude Le AB EBM ge Madan wuulHipttadiow Js vr co wins f tnvaeeee ) * pele mubbipedtion Ts ascot YO GC gutak AB : = . ® BC are fA. Be — A-(® ) defined. gy Nobex Multa plicuktan fs, Dicte\buLry & A. (Becy=aRraAc & ArB)CeACHBC “— A JB,C are contormasele for yespeatve preducks B® AR ore saquore ome OF cane arder . A) IF ABH=HABA AAZBEG gold te'be commute, eo o a z s ! ) oe fa ae (8 fs ||| | ct | | ed] | | NM p L W) ie PBs -BA ALB Sard to be cunt Comune zach oneal ud — Scanned with CamScanner jean : out me 3 :) AE clempote sy omipo “ 12 43 \e pra & a - * ach Zz oR AH o2-2-4 pee AA epte® gre f er VW gy SoAA = Azla 2 4 =w a Ate A-A hs \ 2-3 \ 5 = ine ny 2 (GG 2 4 ) =f 1 g { -2 3 is + pout L + Nilpe r_ Q», art O> re ok yrdaer P 4 A= dent wrod Lair . abwoys qg\wh slpotent Mok ‘s alo “A aXe e \= (: °) 2 eal oe ~ 2 as wee = ohewt Mate) Kok An an uan oa RNS ae fuder BW. a a AS tn * Periodic nade east pre voluc.of ————__ eviedte & | et on & YP . E ATER “alted pened oti mabs%, vs wes above eceurs odte. x Pork A= Yd porow ts pe “ 4 spacmpstent eto pened = A =A=2@ ae ol ron ~ orc A GS 4 : . * Tiweludosyme— yes tg said bo be aw \uvolutery \¢ der we ™™ © watt wos led juvotdery oot oF UM / fb ARedNedn Pian eid “Ax ds call 4] MMaesik or Squave q Mais )a Wodr% \ i > . * AkeD S Att = Ver «\r\=) U a =A pel wr 4 \ar =! Z, hw pryolutory worden is Hs 0 J : afymlex } \wv se | \ Scanned with CamScanner Ceres ee eed bd ddd ddd Ieee I) /)y) ~* Lp Lav\f1 b\ aft eo * fc) mie ‘) () { a { se sng) tos(us) —siv(ve) (ae ) 7 giu(wd) cos(we) tpstue) Stu (ud) ) a Ch - Ty (ATA) = 24 OM On7 + bree +E? He,2e,% =ETYoms pose of oa MotHx ‘ ; = mot abialved Brew A oy Wterelamgi ng its A eclwuw uo” ™ a3)" A= Os = ATS a an (@an ! On ba b) dL bs oe ey Os/an2 eae y ah me olay = 2% otay= 342 ait o(ayemyyt een Ola = new —oon A’ oR AT eh AR FAD . i-e Ci, jPraement oF A = ‘ene dewentoh & * CaryT= A (pB)TS BT AT rod Bo axe como wa ble a CAX BY) 7 AHR (KAY = KAT 4 Ke malay a Scanned with CamScanner Loboboboboobbbed bd bbdd dd ddd ddI CIP YUU E”L. ¢ aif Ovtnogo mah Moree cad matrix tf ¢ OQ three vows / rarer colum , or te eads ohex id ve wokvix \s said to be ortho ae A Layo AAT= IHATA _— L @ a, a3) a be & Oe +05.* +03” Oubstdbtasb, a6 ta,such |e 3 = i Ate by bi bs) ( an lo Cy = | py .dy-bp,t 543 bib itb> be thay = ay b, Cs heyy (GU Cr Ca? 0 +00, 46, biG Holatby, ene Ar +artas =\ Ady +Agb;+ A,bz= S . bir +te,*4+b,> = | bidit bec + bzC =2 Cte ten | Ort Cdsa+Cr3 = © vn are Wt veotors omatiraly ModerX £ oF Acdjoiwt of a Square A = On Are Ors Ae, Ar Ax = =O12 05, O23 & [: A =) co-Factes oF On Cy 22432 23 Asa ‘ G v ad) = (co-Foctor Modi Xx © A) : wy Ged TT rae ody) “ (s ee cs) ie od tA) (<. Cor Coz Ga} Wan Cp—Fastes thotaa | Note: } NOTE: Cox adjolwt oF oem k oh tue she to- Factor oF a, = Cy, = (AY (Ae -453-45,dag Qu Aw Cis Cay Car Ca 4 Car Car Cz =e rf adj (ry eno ano postien % *) -~. P nd ork \wte od -( of mow dtegowal elements Scanned with CamScanner ee 4 Propesties ot Rai jom* ‘ Sepesties OF * A Caan Gy = a* Il a tae)= cain *Yaajim\ = IN a aaj (un) = RO (4 : : a: * ~ nil aaj (at) = (eal ~ \aABl ~ wm \n\ we A is symmede) C teal= ; Veal = ¥ \n . aur ela) te agemebsie 7 "? a gajlodis) IN ® kK ig A \aqgrew aywmedy' cc * \oa taal (N)) ea ucaven Wyeedd een “ —_ : od) (ry V6 ayant, Sc A Vs were czy VSO Lord rg alert \ q i y Ww { shylos | » it aweor= BA bie & B= an n ape G A= & % 1\) A NS \wvest’ RB ts vex marker is whove wad 7 AEA AVpen=AAT ico) * Taverse of O x WANS ox ogo 3 ¥ VER is tvolutory be Ar= AAHE » lal =Al + \EA te wow alugular atiage Az diog Cos Q O5- a) Ke On. n * aon 8,0 Ow 447 - On ‘pre RPERTIES way tay EAS) SAT, Rew Zp * A Souove mal wore Le a, a #0) | A\ - AL * | k i oxtnogeusd an ave av ¥ (As) | BA % aud (A) = A e(adja)" R (RI = Scanned with CamScanner woe NS gonidte PS 7 tp } PotPe IPD 7 Lill 4 Poe Pe if C1 rma CAD iT aT LT iY iD PD PhD Nall a TRA KUM RRURT BAATEIY! Tle te Bd Wl Cy avers aqpare matnt X % Vad as\= \n\ » \ ‘ : aivgio® Son aah} tra) \s si ugura® oA \adi(M =6 - fou Weor em eee + Cad bey Hom eokts Py Ye ou nor er a podyvomial® CA) edrarnc SONVEN NE ev: ? Or) faa) y . ar AC ZO 4 AD _ aseala + Ga Se | won of Product = = \erace Ps ' 3 Qy SX rArTAR enw ot prose Ae eat ot Sy EW Aer ass ae =| o& Doge vb cule Yoken oo oF : Sat dridsds = | Al Awe hemo ; HH Glancntory Mores a gkNns how HPPA O aase Riemontexy Tos * A moet ® obteluet aa) ° “A 4 \ronsPorrradtom ow Wouley ¥s called OM Spe ewnkory word \- . (eer) — (% ° s\-= eure ERT! onteronongina oF omye 0 No a C io my = oo} EREU: mauled plicato wv of arrow oy a seme ¥ \ aren . t&, bona sa{o% ° =|6 16 JERT“ 3° Adding o ecatax mul ie ok a 5 >) meow hs amy otenn TOM? Visas Roo “te (34%) 90) b Ria RRe Scanned with CamScanner TS Tie Te b \ = oO G=(7 5) Met A (% 3 a b (¢ d gaz (es) sb) a wo °o \ b = o A AC, = a »\ 2) a ce | y * mots x A arc * pre. yn VEAGh code) oy awn Planer cowwses Gur We cow? yang to A % Poak wuld plicaor by OM Clomnentary ~ Comey wr ewe Same operakton ov column of A es oa Ger an wows of A Scanned with CamScanner