Section 55 Inverse of a Square Matrix


 Clyde Weaver
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1  Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn in th tal. Playr Dfatd. Ann Dian. Bridgt Ann, Carol, Dian. Carol Ann. Dian Carol, Erln. Erln Ann, Bridgt, Carol (A) Exprss th outoms as an inidn matrix A y plaing a in th ith row and jth olumn of A if playr i dfatd playr j and a othrwis (s Prolm 6). (B) Comput th matrix B A A. (C) Disuss matrix multipliation mthods that an usd to find th sum of th rows in B. Stat th matris that an usd and prform th nssary oprations. (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. Stion  Invrs of a Squar Matrix Idntity Matrix for Multipliation Invrs of a Squar Matrix Appliation: Cryptography In this stion w introdu th idntity matrix and th invrs of a squar matrix. Ths matrix forms, along with matrix multipliation, ar thn usd to solv som systms of quations writtn in matrix form in Stion 6. Idntity Matrix for Multipliation W know that for any ral numr a ()a a() a Th numr is alld th idntity for ral numr multipliation. Dos th st of all matris of a givn dimnsion hav an idntity lmnt for multipliation? That is, if M is an aritrary m n matrix, dos M hav an idntity lmnt I suh that IM MI M? Th answr in gnral is no. Howvr, th st of all squar matris of ordr n (matris with n rows and n olumns) dos hav an idntity. DEFINITION IDENTITY MATRIX Th idntity matrix for multipliation for th st of all squar matris of ordr n is th squar matrix of ordr n, dnotd y I, with s along th prinipal diagonal (from uppr lft ornr to lowr right ornr) and s lswhr.
2 9 SYSTEMS; MATRICES FIGURE Idntity matris. For xampl, and ar th idntity matris for all squar matris of ordr and, rsptivly. Most graphing utilitis hav a uiltin ommand for gnrating th idntity matrix of a givn ordr (s Fig. ). EXAMPLE MATCHED PROBLEM Idntity Matrix Multipliation (A) (B) (C) (D) a d g a d Multiply: (A) (B) h a d 6 a d g f i f h f i a d g a d f a d g a d 6 and h h f 7 8 and 6 f i f i f In gnral, w an show that if M is a squar matrix of ordr n and I is th idntity matrix of ordr n, thn IM MI M If M is an m n matrix that is not squar (m n), thn it is still possil to multiply M on th lft and on th right y an idntity matrix, ut not with th samsiz idntity matrix (s Exampl, parts C and D). To avoid th ompliations involvd with assoiating two diffrnt idntity matris with ah nonsquar matrix, w rstrit our attntion in this stion to squar matris.
3  Invrs of a Squar Matrix 9 Explor/Disuss Th only ral numr solutions to th quation x ar x and x. (A) Show that A satisfis A I, whr I is th idntity. (B) Show that B satisfis B I. (C) Find a matrix with all lmnts nonzro whos squar is th idntity matrix. Invrs of a Squar Matrix In th st of ral numrs, w know that for ah ral numr a, xpt, thr xists a ral numr a suh that a a Th numr a is alld th invrs of th numr a rlativ to multipliation, or th multipliativ invrs of a. For xampl, is th multipliativ invrs of, sin (). W us this ida to dfin th invrs of a squar matrix. DEFINITION INVERSE OF A SQUARE MATRIX If M is a squar matrix of ordr n and if thr xists a matrix M (rad M invrs ) suh that M M MM I thn M is alld th multipliativ invrs of M or, mor simply, th invrs of M. Th multipliativ invrs of a nonzro ral numr a also an writtn as /a. This notation is not usd for matrix invrss. Lt s us Dfinition to find M, if it xists, for W ar looking for suh that M M a d MM M M I
4 9 SYSTEMS; MATRICES Thus, w writ M M I a and try to find a,,, and d so that th produt of M and M is th idntity matrix I. Multiplying M and M on th lft sid, w otain (a ) (a ) d whih is tru only if ( d) ( d) a d a d Solving ths two systms, w find that a,,, and d. Thus, M as is asily hkd: M M I M M Unlik nonzro ral numrs, invrss do not always xist for nonzro squar matris. For xampl, if N thn, proding as for, w ar ld to th systms a d a d Ths systms ar oth inonsistnt and hav no solution. Hn, N dos not xist. Bing al to find invrss, whn thy xist, lads to dirt and simpl solutions to many pratial prolms. In th nxt stion, for xampl, w will show how invrss an usd to solv systms of linar quations. Th mthod outlind aov for finding th invrs, if it xists, gts vry involvd for matris of ordr largr than. Now that w know what w ar looking for, w an us augmntd matris, as in Stion , to mak th pross mor ffiint. Dtails ar illustratd in Exampl.
5  Invrs of a Squar Matrix 9 EXAMPLE Solution Finding an Invrs Find th invrs, if it xists, of M W start as for and writ This is tru only if M M I a d f g h i a d f g h i f h i a d g h Now w writ augmntd matris for ah of th thr systms: First Sond Third Sin ah matrix to th lft of th vrtial ar is th sam, xatly th sam row oprations an usd on ah augmntd matrix to transform it into a rdud form. W an spd up th pross sustantially y omining all thr augmntd matris into th singl augmntd matrix form M I () W now try to prform row oprations on matrix () until w otain a rowquivalnt matrix that looks lik matrix (): I a B d f g h i I B If this an don, thn th nw matrix to th right of th vrtial ar is M! Now lt s try to transform matrix () into a form lik that of matrix (). W follow ()
6 96 SYSTEMS; MATRICES th sam squn of stps as in th solution of linar systms y Gauss Jordan limination (s Stion ): M I ()R R R R R R R R ()R R R R R ( ) R R R R R R Convrting ak to systms of quations quivalnt to our thr original systms (w won t hav to do this stp in prati), w hav a d g h f i And ths ar just th lmnts of M that w ar looking for! Hn, M I B Not that this is th matrix to th right of th vrtial lin in th last augmntd matrix. Chk Sin th dfinition of matrix invrs rquirs that M M I and MM I () it appars that w must omput oth M M and MM to hk our work. Howvr, it an shown that if on of th quations in () is satisfid, thn th othr
7  Invrs of a Squar Matrix 97 MATCHED PROBLEM is also satisfid. Thus, for hking purposs it is suffiint to omput ithr M M or MM w don t nd to do oth. M M Lt M (A) Form th augmntd matrix M I. (B) Us row oprations to transform M I into I B. (C) Vrify y multipliation that B M. I Th produr usd in Exampl an usd to find th invrs of any squar matrix, if th invrs xists, and will also indiat whn th invrs dos not xist. Ths idas ar summarizd in Thorm. THEOREM INVERSE OF A SQUARE MATRIX M If M I is transformd y row oprations into I B, thn th rsulting matrix B is M. If, howvr, w otain all s in on or mor rows to th lft of th vrtial lin, thn M dos not xist. Explor/Disuss (A) Suppos that th squar matrix M has a row of all zros. Explain why M has no invrs. (B) Suppos that th squar matrix M has a olumn of all zros. Explain why M has no invrs. EXAMPLE Finding a Matrix Invrs Find M, givn M 6 Solution R R 6 6 6R R R R R
8 98 SYSTEMS; MATRICES Thus, R R R M Chk y showing M M I. MATCHED PROBLEM Find M, givn M 6 EXAMPLE Solution Finding an Invrs Find M, if it xists, givn M W hav all s in th sond row to th lft of th vrtial lin. Thrfor, M dos not xist. MATCHED PROBLEM Find M, if it xists, givn M 6 Most graphing utilitis an omput matrix invrss and an idntify thos matris that do not hav invrss. A matrix that dos not hav an invrs is oftn rfrrd to as a singular matrix. Figur illustrats th produr on a graphing utility. Not that th invrs opration is prformd y prssing th x ky. Entring [A]^() rsults in an rror mssag. FIGURE Finding matrix invrss on a graphing utility. (a) Exampl () Exampl
9 Appliation: Cryptography  Invrs of a Squar Matrix 99 Matrix invrss an usd to provid a simpl and fftiv produr for noding and doding mssags. To gin, w assign th numrs to 6 to th lttrs in th alphat, as shown low. W also assign th numr 7 to a lank to provid for spa twn words. (A mor sophistiatd od ould inlud oth uppras and lowras lttrs and puntuation symols.) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Blank Thus, th mssag I LOVE MATH orrsponds to th squn Any matrix whos lmnts ar positiv intgrs and whos invrs xists an usd as an noding matrix. For xampl, to us th matrix A to nod th aov mssag, first w divid th numrs in th squn into groups of and us ths groups as th olumns of a matrix B with two rows: B Prod down th olumns, not aross th rows. (Noti that w addd an xtra lank at th nd of th mssag to mak th olumns om out vn.) Thn w multiply this matrix on th lft y A: AB Th odd mssag is This mssag an dodd simply y putting it ak into matrix form and multiplying on th lft y th doding matrix A. Sin A is asily dtrmind if A is known, th noding matrix A is th only ky ndd to dod mssags nodd in this mannr. Although simpl in onpt, ods of this typ an vry diffiult to rak.
10 SYSTEMS; MATRICES EXAMPLE Cryptography Th mssag Solution was nodd with th matrix A shown low. Dod this mssag. A W gin y ntring th noding matrix A (Fig. ). Thn w ntr th odd mssag in th olumns of a matrix C with thr rows (Fig. ). If B is th matrix ontaining th unodd mssag, thn B and C ar rlatd y C AB. To find B, w multiply oth sids of th quation C AB y A (Fig. ). FIGURE FIGURE Writing th numrs in th olumns of this matrix in squn and using th orrspondn twn numrs and lttrs notd arlir produs th dodd mssag: W H O I S C A R L G A U S S Th answr to this qustion an found arlir in this haptr. MATCHED PROBLEM Th mssag was nodd with th matrix A shown low. Dod this mssag. A
11  Invrs of a Squar Matrix Answrs to Mathd Prolms. (A) (B) (A) (B) (C).. Dos not xist. WHO IS WILHELM JORDAN EXERCISE  A Prform th indiatd oprations in Prolms In Prolms 9 8, xamin th produt of th two matris to dtrmin if ah is th invrs of th othr. 9.. ;.. ; ; ; 7 ; ; B Givn M in Prolms 9 8, find M, and show that M M I ; ; ; ; Find th invrs of ah matrix in Prolms 9, if it xists
12 SYSTEMS; MATRICES.. C Find th invrs of ah matrix in Prolms 8, if it xists Disuss th xistn of M for diagonal matris of th form. Disuss th xistn of M for uppr triangular matris of th form. Find A and A for ah of th following matris. (A) (B) A A. Basd on your osrvations in Prolm, if A A for a squar matrix A, what is A? Giv a mathmatial argumnt to support your onlusion.. Find (A ) for ah of th following matris. (A) M a M a A (B). Basd on your osrvations in Prolm, if A xists for a squar matrix A, what is (A )? Giv a mathmatial argumnt to support your onlusion.. Find (AB), A B, and B A for ah of th following pairs of matris. (A) A (B) A d d A and B and B Basd on your osrvations in Prolm, whih of th following is a tru statmnt? Giv a mathmatial argumnt to support your onlusion. (A) (AB) A B (B) (AB) B A APPLICATIONS Prolms 7 rfr to th noding matrix 7. Cryptography. Enod th mssag CAT IN THE HAT with th matrix A givn aov. 8. Cryptography. Enod th mssag FOX IN SOCKS with th matrix A givn aov. 9. Cryptography. Th following mssag was nodd with th matrix A givn aov. Dod this mssag Cryptography. Th following mssag was nodd with th matrix A givn aov. Dod this mssag Prolms rfr to th noding matrix A B. Cryptography. Enod th mssag DWIGHT DAVID EISENHOWER with th matrix B givn aov.. Cryptography. Enod th mssag JOHN FITZGER ALD KENNEDY with th matrix B givn aov.. Cryptography. Th following mssag was nodd with th matrix B givn aov. Dod this mssag Cryptography. Th following mssag was nodd with th matrix B givn aov. Dod this mssag
SECTION 92 Inverse of a Square Matrix
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