Matrix Algebra Notes

Transcription

1 Sectio About these otes These re otes o mtrix lgebr tht I hve writte up for use i differet courses tht I tech, to be prescribed either s refreshers, mi redig, supplemets, or bckgroud redigs. These courses re primrily Itermedite Mth for Ecoomics Ecoomic Forecstig Advced Mthemticl Methods vrious Ecoometrics courses Perhps others will fid them useful too. They were desiged for self-study, d iteded s quick d dirty itroductio to the essetils i mtrix lgebr. If you re usig this, plese ote: - The exercises re essetil prt of the otes. My of the importt cocepts d results re i the exercises. Do the exercises. - These otes re ot met to replce y textbooks o mtrix lgebr. Plese use these oly s supplemet to the rel textbooks. - The idividul sectios re ot std-loe. Nor complete i its expositio; I hve mde umber of ssumptios regrdig wht you lredy kow. - I cll these otes to mtrix lgebr s opposed to lier lgebr, which gets ito vector spces. The coverge i these otes is terribly ive i compriso. Plese see Strg (009) or tke rel Lier Algebr course. - These otes re icomplete. There re my topics I pl to dd, d will do so from time to time, with o prticulr sese of urgecy. Athoy Ty Sigpore Mgemet Uiversity Athoy Ty -

2 Sectio Systems of Equtios Solvig ecoomic models ofte requires solvig systems of equtios. Here we discuss solvig systems of lier equtios, meig tht the equtios i the system tke the form x x x c. Solutio Possibilities (i) x x 4 x x... Cosider the system A solutio of this system is pir of umbers ( x,) x tht stisfy both equtios t the sme time. I this exmple, the solutio is esily foud to be ( x,)(,0) x. The system (i) hs exctly oe solutio, but this is ot true for ll systems. Some hve ifiite umber of solutios, others hve oe. Cosider (ii) 3x 5x 6 6x 0x The secod equtio is simply two times the first they re ot idepedet equtios. There is effectively oly oe equtio i two ukows, d y ( x,) x tht stisfies the first equtio will stisfy the secod. The system hs ifiitely my solutios y ( x,) x stisfyig x (5/ 3) x or x (6/5)(3/ 5) x is solutio. The set of solutios is usully writte s 5s ( x,) x, 3 s or x x The symbol s is sometimes clled prmeter. 6 3s (,), s 5 5. The fct tht system my hve o solutio is illustrted by the ext exmple: (iii) 3x 5x 6 3x 5x 7 If the first equtio is stisfied, the secod cot be stisfied. This system hs o solutio. The two equtios re coflictig equtios. Drwig the equtios o the x-y ple (more ppropritely here, the x - x ple) shows clerly wht hppes i ll three situtios. The two lies itersect i (i), coicide i (ii), d re prllel to ech other i (iii). Athoy Ty -

3 A system with t lest oe solutio is clled cosistet system. A system with o solutio is sid to be icosistet. Whether system hs oe solutio or my solutios depeds o whether there re s my idepedet equtios s there re vribles to be solved. Wht if there re three equtios i two vribles? Here we hve ll three possibilities. Cosider (iv) x 3x 8 3x 4x 7 x x 0 This system hs o solutio. Solvig the lst two oly gives the uique solutio x 3/ 5, x 37 /0. But this solutio cot be recociled with the first equtio: x 3x (3/ 5) 3(37 /0) 63/0 8. Drwig the three equtios i the x - x ple shows clerly wht is goig o here. For there to be solutio, these three equtios must itersect t the sme poit. I this exmple, they do t. Here is exmple where the three equtios do itersect t sigle poit. (v) 4x x 3 3x 4x 7 x x 0 System (v) hs exctly oe solutio. Solvig the lst two gives x 3/ 5, x 37 /0 s i the previous cse. Substitutig this solutio ito the first equtio gives 4x x 4(3/ 5) (37 /0) 5/ 5 3 so the first equtio is lso stisfied. Aother wy of lookig t this system is to observe tht lthough there re three equtios i two ukows, there re ctully oly two idepedet equtios. Ay oe of the three equtios c be writte s combitio of the other two (e.g., 3rd equtio is st mius d; or d equtio is st mius 3rd, etc.) Solvig y two therefore solves the third. It is the umber of idepedet equtios tht mtter i determiig the umber of solutios: two idepedet equtios i two ukows produces sigle solutio. Athoy Ty -

4 If three-equtio two-vrible system cotis oly oe idepedet equtio, the effectively we will hve oly oe equtio i two ukows, ledig to ifiitely my solutios. This is the cse with the followig system: (vi) x 4x 0 3x 6x 30 x x 0 Here the st equtio is twice tht of the 3rd, d the secod is three times tht of the 3rd. Therefore there is oly oe idepedet equtio, i two ukows. There re ifiitely my solutios ( x,)(0 x,) s s. ******* A lier equtio ivolvig three vribles x, x, d x 3, represets ple i the 3-dimesiol spce. This visuliztio helps you to uderstd wht c hppe i lier systems with three vribles. If you hve two ples, the either the ples re prllel, re coicidet, or itersect. If they itersect, the itersectio produces lie i three-dimesiol spce. Therefore two lier equtios ivolvig three vribles will either hve o solutios (if the two ples re prllel), or ifiite umber of solutios represeted by the etire ple (if the two ples coicide) or ifiite umber of solutios represeted by the lie of itersectio of the two ples. The followig system hs ifiitely my solutios represeted by sigle lie. (vii) x 3x x 0 3 x x x 3 Subtrctig the secod from the first elimites x 3, d gives x 4x. Thus for y x s, we hve x 4s. Substitutig ito the secod equtio gives x x x 4s s 5s. 3 We hve ifiitely my solutios give by ( x, x,)(4 x3, s, 5) s s. The system (viii) x 3x x 0 3 4x 6x x 3 however, hs o solutio. Dividig the secod equtio by gives x 3x x3 /. Thus if ( x, x,) x3 stisfies the first, it cot stisfy the secod. These two ples re prllel. Athoy Ty -3

5 The system (ix) x 3x x 3 4x 6x x 3 hs ifiitely my solutios. It is esy to see tht the two equtios re ideticl. Every poit o the ple is solutio. For y x s, x t, we hve x3 s 3t. The solutio is ( x, x,)( x,, s t 3) s t. 3 ******* I order to get uique solutio to three-equtio system, there must be three idepedet o-coflictig equtios. The three ples must itersect t sigle poit. If there re y depedece mog the three equtios, the we will ed up with ifiitely my solutios (either lie or ple), or if there re coflictig equtios, o solutios. We will explore this further i lter sectios. Solvig lrger systems of equtios my seem more dutig tsk, but the process of solvig smll systems is esily systemtized. This process is clled elimitio usig elemetry row opertios. Tke the system: x x 4x 4 eq 3 3x x x eq 3 5x x x 7 eq3 3 You would geerlly proceed by usig equtio to elimite x from eq d eq3: x x 4x 4 eq 3 eq mius 3 / eq 0x x 4x3 4 eq eq3 mius 5 / eq 0x 3x 9x3 3 eq3 The use eq to elimite x from eq3, to get the trigulr system x x 4x 4 eq 3 0x x 4x 4 eq 3 eq3 mius 3 / eq 0x 0x 3 x3 3 eq3 Athoy Ty -4

6 Filly, multiply the third equtio by / 3 to get x3. Substitutig x3 upwrds gives x 4, d x 0. You my hve to switch the positios of your equtios first. If the coefficiet o the first vrible i the first equtio is zero, the switch tht equtio with other: 0x x 4x3 4 3x x x3 3x x x3 switch eq with eq 0x x 4x3 4 5x x x3 7 5x x x3 7 d proceed s usul. Notice the importt role plyed by the boxed umbers. These umbers re clled the pivots of the coefficiet mtrix. If system of equtios hs uique solutio it will hve o-zero pivots. Notice lso tht we oly used three types of opertios i solvig our system of equtios: switchig equtios, multiplyig by (o -zero) costt, d usig oe equtio to elimite other. These opertios re clled row-opertios d uderlie much of mtrix lgebr. Systems of Nolier Equtios ******** We ofte lso hve to solve systems of o-lier equtios, such s (x) xy 4 x y 8 The method for solvig systems of o-lier equtios is effectively lso to elimite vribles, lthough how best to do this depeds of the system; it is cseby-cse situtio. For the system bove, we might pproch it i the followig wy: the first equtio gives x 4 / y. Thus which gives 6 / 8 y y ( 4) 0 y. 6 y 8y 4 The solutios re, therefore, ( x,)(,) y d ( x,)( y, ). Athoy Ty -5

7 It is esy to see tht solvig lrger systems of o-lier equtios c get relly tricky. Eve smll systems c be tricky, d you ll hve to be creful of extreous solutios d missig solutios. For exmple, tke (xi) y x y x Perhps the obvious thig to do here is x x x () x 4 4 x x 0( 4)( ) x x. So you might coclude tht x or x 4, with y d y respectively. However, ( x,)(4,) y is ot solutio, sice this does ot lie o the secod equtio. [Q: why is ( x,)(4, y ) ot solutio?] Exercises. Plot systems (i) to (vi), d (x) d (xi), ech i their ow digrms.. Solve the system y x x y, for x 0, y 0, 0. Plot the system o the x - y ple for differet vlues of. Athoy Ty -6

8 Sectio 3 The Summtio Nottio You will frequetly del with complicted expressios ivolvig lrge umber of dditios. Ofte, these expressios re simplified usig the summtio ottio. My studets fid difficulty i mipultig such expressios. The purpose of this sectio is to itroduce the ottio to you, d to get you comfortble with it. 3. Defiitio d Rules for Sums The uppercse sigm is used to deote summtio. For rbitrry set of umbers { x, x,..., x } defie xi x x... x. i Exmple 3.. The verge of set of umbers x, x,..., x c be writte x x i i. Exmple 3.. Write the sum i summtio ottio: 6 As: i. 4 i Exmple 3..3 Suppose the followig pymets re to be mde: i the first period, i the secod period, d so o util i the th period. At fixed iterest rte of r per period, the preset vlue of the pymets is i.... i r ()()() r r r i I Exmple 3.., the idex of summtio i eter s subscripts, but otherwise do ot eter ito the computtio of the terms of the summtio. I Exmple 3.., the idex is ctully prt of the computtio of the terms. I Exmple 3..3, the idex is used both wys. Athoy Ty 3-

9 Exmple 3..4 Ecoomists ofte use ggregte price idex to trck the overll price level i ecoomy reltive to some bse yer. This is usully doe by trckig weighted verge of prices of certi set of commodities. Let i,..., q0i represet commodities be the qutity of good i purchsed i period 0 (the bse yer) p0i be the price of good i i period 0 qti pti The Lspeyres Price Idex is be the qutity of good i purchsed i period t be the price of good i i period t i i p q ti p q 0i 0i 0i p tiq i The Psche Price Idex is p q i 0i ti ti Expressios usig summtio ottio re ot uique; more th oe expressio c be used to represet give sum. Exmple 3..5 Write /3 + /5 /7 + /9 / i summtio ottio: As: i. Alterte swer: 6 i ( ) i 5 i0 ( ) i i 3. Rules for Workig with the Summtio Nottio The summtio ottio gretly simplifies ottio (oce you get used to it), but this is oly helpful the you kow how to mipulte expressios writte i it. There re oly two rules to ler (i) () b b, (ii) i i i i i i i Exmple 3.. i c c ci c i, where c is costt. i i Exmple 3.. i () x x0 i, where x xi i. Proof: () x x x x x x 0 i i i i i Athoy Ty 3-

10 Exmple 3..3 ()()()() x x y y x x y y y x, i i i i i i i i i where x xi i, d y yi i Proof We hve ()()()() x x y y x x y x x y i i i i i i i i But the secod term is zero: which gives the desired result. ()() x x 0y y xi x, i i i The proof of ()()() x x y y y y x is similr. i i i i i i 3.3 Some useful formuls ivolvig summtios For every iteger, i i 3... ( ) i i 3... ( )( ) i 3... i i i Arithmetic Series ()(( id )) i d i0 i ()( d )...(( d )) d ( ) d where d d re rel umbers. Geometric Series () r r r r r r () r, i i... i0 i where d r re rel umbers. Athoy Ty 3-3

11 3.4 Double summtios Suppose we hve rectgulr rry of umbers m m m Let the totl sum of these umbers be S. To get S we c first dd up the rows d the dd the results, m i.e., S j j m j ij, j j j i j or first dd up the colums, m m m m i.e., S i i i ij. i i i j i The prethesis mkes it cler which summtio is to be doe first, but it is covetiol to leve out the prethesis d write m ij i j or m j i ij with the uderstdig tht the summtios re crried out from right to left, i.e., from the ier summtio to the outer. Exmple 3.4. Expd m i j. i j Oe wy to do this is m m i j i j ( )( m) i j i j A more explicit rgumet i j i j i j i j m m m m i j i j i j i j ( )( ) m Athoy Ty 3-4

12 I the exmples so fr, we could iterchge the summtio sigs, i.e., m = ij i j m ij. j i We c do this oly if the limits of the outer summtio do ot deped o the limit of y of the ier summtios. Exmple 3.4. Suppose we hve trigulr rry of umbers to be expressed i summtio ottio. m i m m mm We c write i j ij. We cot iterchge the summtio symbols becuse the upper limit i the ier summtio depeds o the idex of the outer summtio. Exmple ottio. where m. Write the sum of the followig trigulr rry usig summtio,,, m, m, m, Solutio: Let i, j be the typicl elemet i the sum. The the first colum hs j, d i ruig from to m ; the secod colum hs j, d i ruig from to m. I geerl we hve j ruig from to, for the j th colum, i ruig m from j to m. Thus the sum is. j i j ij Athoy Ty 3-5

13 Exercises. Write out i full, the evlute or simplify. e. i i b. 4 0 i (i ) f. 3 i0 0 ix c. i ( ) i i g. 4 ( i ) x i d. i i j j h. 0 i i j i. Write i summtio ottio b. m m m m b b b b m m m m m m m... c. + 3/ + 4/3 + 5/4 + 6/5 d. b b... b i j i j i j 3. Prove by writig out i full. c. e k [( k ) k ] b. 3 3 i j i j j i i j x x x x j i j i j j x x x x d. 3 3 x i x j x i j i i x j j x x x x where i i i i 4. Prove usig the rules of summtio:. i i i b. i j i 3 3 i x xi j i ji i i 5. Prove ll the equlity reltios i the followig: ()() xi x xi x xi xi x i i i 6. Show tht ()() x x y y x y x y. i i i i i i Athoy Ty 3-6

14 7. Evlute by first simplifyig, the pplyig the pproprite formuls. 30 k k( k ) b. k k( k )( k ) 8. Let { x, x,..., x0 } {,, 6,,, e, 0,,, 4. }. Verify usig excel or 3 otherwise (clcultor, mul computtio, metl computtio, s you plese) tht. b. c. 0 () x 0 i x i 0 () 0 x 0 i x x i i i x 0 0 ()() x i x x i i i x xi If i (i) you get swer little differet from 0, expli why this occurs. 9. Express the followig i the form x bx cx 3 dx 4 : (i) 4 i j i x j (ii) 4 4 (i.e., you hve to tell me wht, b, c, d d re i ech cse.) i ji i x j 0. Let { x, x,..., x } be rbitrry set of rel umbers, d let x i i x. Prove tht. ()( x )()( 0000) i x x i i x i i x x i Athoy Ty 3-7

15 Sectio 4 Mtrix Algebr: Defiitios d Bsic Opertios Defiitios Alyzig ecoomic models ofte ivolve workig with lrge sets of lier equtios. Mtrix lgebr provides set of tools for delig with such objects. A mtrix is rectgulr collectio of umbers A. m m m The umber of rows m eed ot be equl to the umber of colums. A mtrix with m rows d colums is sid to hve order (m,) or dimesio (m,), or we simply cll it ( m ) mtrix. The umber tht ppers i the (i, j)th positio is clled the (i, j)th elemet or the (i, j)th etry of the mtrix. If m, the mtrix is squre mtrix. If m d, it is clled row vector. If m d, we hve colum vector. If m, the we hve sclr. The elemets of vector re ofte clled the compoets of the vector. b b Exmple A row vector c c c c, colum vector b. bm By simply sttig tht b is vector we will usully me tht b is colum vector, but you eed to be wre whether row or colum vector is beig referred to. Mtrices re ofte writte i uitlicized bold uppercse letters, vectors i uitlicized bold lowercse letters. It is ofte coveiet to write mtrix s A ( ), ij m d ofte coveiet to refer to the (i, j)th elemet of A usig ij A. There is vritio i ottio from uthor to uthor, so be creful i your redig. Two mtrices of the sme dimesio m re sid to be equl if ll of their correspodig elemets re equl, i.e., A B [ A] ij [ B ] ij i,,..., m, j,,...,. Mtrices of differet dimesios cot be equl. Athoy Ty 4-

16 Bsic Opertios (Additio, Sclr Multiplictio, Subtrctio, Trspose) Additio Let A ( ) d B ( b ) be two rbitrry ( m ) mtrices. Defie ij m ij m A B ( ij bij ) m, i.e., dditio of mtrices is defied to be elemet by elemet dditio. Exmple Mtrices beig dded together obviously must hve the sme dimesios. It should lso be obvious tht A B B A ( A B) C A ( B C ) This mes tht s fr s dditio is cocered, we c mipulte mtrices i the sme wy we mipulte ordiry umbers (s log s they hve the sme dimesios) Sclr Multiplictio Let A be ( m ) mtrix, d be sclr. The defie A ( ij ) m i.e., the product of sclr d mtrix is defied to be the multiplictio of ech elemet of the mtrix by the sclr. Exmple b b b b b 3 3 b3 b3 We c use sclr multiplictio to defie mtrix subtrctio. Let A d B be ( m ) mtrices. The A B A ( ) B. Exmple Athoy Ty 4-

17 Trspose A importt opertor is the trspose. Whe we trspose mtrix, we write its rows s its colum, d its colums s its rows. For exmple, deotig the trspose of A by A we hve Put more succictly, [ A ] ij [ A ] ji. Note tht trsposes re ofte deoted usig A isted of A. Oe pplictio of the trspose opertor is i defiig symmetric mtrices. A symmetric mtrix is defied s oe where A A. Exercises. Let 7 3 A 4 4. Wht is the dimesio of A? Wht is [ A ]? Wht is [ A ] 3? 7 5. Suppose A ( ij ) 4 where ij i j. Write out the mtrix i full. 3. Write out i full the mtrix (i) ( ij ) 4 4 where ij whe i j, 0 otherwise. (ii) ( ij ) 4 4 where ij 0 if i j. (Fill i the rest of the etries * ) (iii) ( ij ) 55 where ij 0 whe i j. (Fill i the rest of the etries with * ) (vi) ( ij ) 55 where ij 0 whe i j. (Fill i the rest of the etries with * ) These re ll squre mtrices. Mtrices (i) d (ii) re clled digol mtrices. Mtrix (iii) is lower trigulr mtrix, d (iv) is upper trigulr mtrix (so we hve i (iii) d (iv) mtrices tht re squre d trigulr!) 4. Give exmple of (4 4) mtrix such tht [ A] [ A ]. ij ji 5. If u v u v, wht is u d v? Athoy Ty 4-3

18 6. Let v, v, v 3, v 4 represet cities, d suppose there re oe-wy flights from v to v d v 3, from v to v 3 d v 4, d two-wy flights betwee v d v 4. Write out mtrix A such tht [ A ] ij if there is flight from v i to v j, d zero otherwise. 7. Wht is the dimesio of the mtrix ? Let A d B 0 0. Is A B? 0 0 Mtrices with ll zero etries re clled zero mtrices, d writte 0 m,, or 0 if squre, or simply 0 if the dimesios c be esily obtied from cotext. 9. If u , wht is u, v, d w? 5 v w 7 0. If 3 4 A 8, wht is A? If B 8 5, wht is B? 4 3. Which of the followig mtrices re symmetric? () b b 3 (b) b (c) (d) b b 3 (e) Athoy Ty 4-4

19 . True or Flse? () Symmetric mtrices must be squre. (b) A sclr is symmetric. (c) If A is symmetric, the A is symmetric. (d) The sum of symmetric mtrices is symmetric. (e) If ( A ) A, the A is symmetric. 3. () Fid A d B if they stisfy simulteously. AB AB 5 (b) If A B C d 3A B 0 simulteously, fid A d B i terms of C. Athoy Ty 4-5

20 Sectio 5 Mtrix Algebr: Multiplictio Let A be ()m mtrix d B be () p. These dimesios c be rbitrry, but the umber of colums of A d the umber of rows of B must be the sme. The product AB is defied s the ()m p mtrix whose ( i,) j th elemet is defied by i j k AB b. Tht is, the ( i,) j th elemet of the product AB is defied s the sum of the product of the elemets of the i th row of A with the correspodig elemets i the th j colum of B. For exmple, the (,) th elemet of AB is The (,3)th elemet is AB kbk b b 3b3... b, k AB kbk3 b3 b3 3b33... b3,3 k Visully, for product of (3 3) mtrix ito (3 ) 3 b b b b 3b3 3 b b b3 b 3 3 b b b b 3b3 b b 3b 3 3 b b b3 b 3 ik k j mtrix, we hve 3 b b b b 3b3 b b 3b3 3 b b b b 3b3 b b 3 d so o... Exmple 8 Let A d B 6 9. The 5 AB 8()(4)(8)(6)()(7)(8)(9) (3)(4)(0)(6)(3)(7)(0)(9) (5)(4)()(6)(5)(7)()(9) 6 44 Exmple Let A 0 4 d B 5. The 0 AB (6)(4)(5)(5)( )(0)(6)( )(5)()( )() ()(4)(0)(5)(4)(0)()( )(0)()(4)() Athoy Ty 5-

21 Exmple The simulteous equtios x x 4 x x c be writte i mtrix form by defiig we c write the system s A, x x x, d x 4 x, or simply Ax b. 4 b. The Exercises My of these exercises provide more th just prctice with mtrix multiplictio; my of the exercises illustrte properties of mtrix multiplictio which differ from multiplictio mog ordiry umbers. As you do these exercises, sk yourself wht the lesso is.. Fid AB whe () 0 0 A d 8 0 B (b) 5 A 0 4 d 4 B 5. Let 8 A 3 0, 5 0 B 3 8, d 7 C 6 3 (i) Compute BC ; (ii) Compute CB ; (iii) C BA be computed? Remrk: This exercise shows tht for y two mtrices A d B, AB BA. We distiguish betwee premultiplictio d postmultiplictio. I the product AB, we sy tht B is premultiplied by A, or A is postmultiplied by B. 3. Let 3 d d 4 6 f. Compute (i) fd (ii) df (iii) d d. Athoy Ty 5-

22 4. Show tht for y vector x x x, the product x x 0. Whe will x x 0? x 5. (i) Compute 4 4 (ii) Let A b b. Compute A b b AA b b Remrk A mtrix with ll elemets equl zero is clled the zero mtrix 0. Obviously, A 0 A d A0 0A 0. Note however tht AB 0 does ot imply A 0 or B 0. I fct, the squre of o-zero mtrix c be zero mtrix! Mtrix multiplictio therefore does ot behve like the usul multiplictio of umbers: the order of multiplictio is importt, d AB 0 does ot imply tht either A 0 or B 0. Mtrix multiplictio, however, does follow ssocitive d distributive lws: (AB)C = A(BC) A(B + C) = AB + AC (A + B)C = AC + BC These re esy to prove. 6. Compute (i) (ii) Remrk The squre mtrix I Athoy Ty 5-3

23 is clled the idetity mtrix. It behves like the umber i regulr multiplictio: for y mtrix A, AI IA A. To emphsize the dimesio of idetity mtrix, we sometimes write I if it is (). 7. Compute (i) b b b (ii) b b b33 Remrk: Mtrices B where 0 mtrices. (iii) 3 b 3 b b B for ll i j i j (iv) b b b b 3 4 re clled digol Show tht 3 3 b 3 3 b b b b b Show tht 3 3 b b b3 b b b b b Write the simulteous equtios 4x z 4 9x y 3z 3 7x y i mtrix ottio. Athoy Ty 5-4

24 b b b3 3. For A 4 5 d b4 b5 b B 6, prove tht () AB B A by 6 b7 b8 b 9 multiplyig out the mtrices. Remrk: This result holds geerlly. For y ()m () p mtrix B, we hve () AB mtrix A d y B A. The proof is ot difficult, d provides very good exercise i usig the ottio we developed erlier. We wt to show tht the ( i,)th j elemet of () AB elemet of () AB B A. By the defiitio of the trspose, ( i,)th j is the ( j,)th i elemet of AB, therefore is equl to the ( i,)th j elemet of [() AB] [ ] AB i j k ji jk k i k b b k i jk B ik A k j k B A i j.. Prove tht () ABC C B A. 3. Let X be geerl () k X X is symmetric. mtrix. Expli why X X is squre. Expli why 4. Give () squre mtrix A, defie the trce of the mtrix to be i ii trce() A Tht is, the trce of squre mtrix is simply the sum of its digol elemets. Show tht trce()()() A B trce A trce B (of course, both mtrices must be the sme size) trce()() A trce A trce() ABtrce() BA (here A d B eed ot be of the sme size). Hit for the lst oe: look t the proof give i questio d dpt it. Athoy Ty 5-5

25 Sectio 6 Itroductio to the Iverse Mtrix The iverse of squre mtrix A is the mtrix, deoted by A A I. A, such tht Exmple sice The iverse of the mtrix A is A 5, A A Oe pplictio of mtrix iverses is i solvig simulteous equtios. Tke for exmple which c be writte i mtrix form s x x 4 x x Ax b, where A, x x x, d 4 b. Sice we kow A, we c simply (pre)multiply Ax b with A Ax A b. A to get Sice A A I, d Ix x, we hve x A b. This is the solutio to the system. 4 A b 5 0 You c verify o your ow tht x d x 0 solves the equtios. The formul for the iverse of rbitrry ( ) mtrix is A Athoy Ty 6-

26 A A To show this, multiply the two together: where A. A A A 0 A 0 A. 0 0 The formul for the iverse of ( ) mtrix is worth committig to memory. The expressio A is clled the determit of A, somethig we will discuss i detil i lter sectios. Note tht if A 0, the the iverse will ot exist (we sy tht A is sigulr ). If A is sigulr, the system will ot hve uique solutio. Exercises. Fid the iverse of the followig mtrices (i) (ii) 3 4 (iii) 4 (iv) 0 0 (v) 0 0 (vi) (vii) (viii) 3 (ix) 0 0 (x) 7 4 (xi) 0 0 (xii) 0 0. Fid the iverse of the mtrix 0 Verify by directly multiplictio. 0 d. Athoy Ty 6-

27 3. Mke guess s to the iverse of the () mtrix Verify your cojecture by directly multiplictio. 4. Fid the iverse of the mtrix 0 b c Solve the followig systems of equtios by computig the iverse of the coefficiet mtrix: (i) (iv) x x 4 x x y x 4 y x (ii) (v) 3x 5x 6 6x 0x x 3x 0 x x 0 (vi) (iii) 3x 5x 6 6x 0x 0 3x 5x 0 6x 0x 0 6. Let 3 A 4, B.5, d C 3. (i) Show tht AB AC. (ii) Show tht A does ot exist. Remrk The exmple i (i) shows tht AB AC does ot imply tht B C i geerl. However, if A exists, the it must be tht B C, sice AB AC A AB A AC B C. 7. We defied the iverse mtrix of A s the mtrix 8. Let Show tht this implies tht AA premultiplyig or postmultiplyig A with A c b d. Fid the iverse of iverse exists), d show tht A such tht A A I. I. Tht is, it does t mtter whether you A, you still get I s result. A (ssume, b, c, d re such tht the ()() A A. Athoy Ty 6-3

28 ()() 9. Let A be () mtrix whose iverse exists. Show tht A A. (You do t eed to kow how to compute () for this. Strt with the fct tht A A I d tke trsposes. 0. Let A Show tht d b B b b b, d ssume tht their iverses exist. () AB B A.. Let A d B be () mtrices whose iverses exist. Show tht () AB B A. 0. Let A 0 3 d B 0. Fid the iverse of AB. Does the reltioship () AB B A hold for these two mtrices? Why? 3. Is it true tht () A B A B? Give couterexmple. Athoy Ty 6-4

29 Sectio 7 Fidig Iverse usig Elemetry Row Opertios The formul for the iverse of (33) d lrger squre mtrices is much more complicted. We will see them i lter sectio. For ow, we show prcticl (but tedious) wy to fid the iverse of mtrix usig elemetry row opertios. These opertios re exctly the steps used i the elimitio process for solvig systems of equtios. The ide here is tht the three elemetry row opertors: Switchig rows Subtrctig costt times oe row from other row Multiplyig etire row by some umber c ll be replicted by premultiplictio by certi mtrices. For exmple, to switch rows d i the mtrix 0 4 A 3 6 we c premultiply by 0 0 E which is obtied by switchig the st d d rows of the (3 3) idetity mtrix. You c verify tht premultiplyig A by E switches rows d of A : EA Similrly, premultiplyig mtrix by 0 0 E ()(3) 0 0, 0 0 which is obtied by switchig the d d 3 rd rows of the (3 3) idetity mtrix, switches the first d third rows of the mtrix: This geerlizes to switchig other pirs of rows, d to lrger mtrices. Athoy Ty 7-

30 Aother type of elemetry row opertio is to dd/subtrct costt times oe row to other row. For exmple, you c use the 3 i row s pivot to elimite the 5 below it usig this opertio. 3 3 e.g. row = row row d use the resultig 3 i row s pivot to elimite the i row 3: e.g. row3 = row3 6 row These opertios c lso be doe by premultiplyig with pproprite mtrices. To subtrct 5 of row from row, premultiply the mtrix by which is obtied by tkig idetity mtrix d subtrctig 5 of row from row. 3 Verify tht To subtrct 6 row from row 3, premultiply by which is obtied by tkig idetity mtrix d subtrctig 6 row from row. Verify tht The ptter should be cler: to fid the pproprite mtrix for executig y prticulr row opertio, tke the idetity mtrix d pply tht sme opertio to it. Athoy Ty 7-

31 To multiple row three by, premultiply by We hve If the iverse of mtrix A exists, the there is series of elemetry row opertors tht reduces A to the idetity mtrix. Suppose the required elemetry row opertors re (i order) E, E,..., E, the E... E E A I which mes tht E... EE A. Furthermore, becuse E... EEI E... EE, we c use the followig techique: Write A d I side-by-side. The pply the sme row opertors to both A d I util A is reduced to the idetity mtrix. At the sme time, the idetity mtrix will be reduced to the iverse mtrix. A I EA EI EEA EEI E... EEA E... EEI reduced to I A Here is the fully worked out exmple for our exmple mtrix: switch rows d row 3 mius row, Athoy Ty 7-3

32 4 row mius row 3 3 row mius row row mius row row row row 3 3 The iverse of A 3 is therefore You should verify this by computig or d checkig to see if you get the idetity mtrix. If we re usig this techique to solve the equtio Ax b, i.e. to compute x A b : write dow A d b side-by-side d pply the row opertios to redure A ito the idetity mtrix. The rtiole for this is the sme s previously: A b EA Eb EEA EEb E... EEA E... EEb reduced to I A b Athoy Ty 7-4

33 Exercises. Fid the iverse of the mtrix 3 A The mtrix A 3 hs o iverse. Apply elemetry row opertios 6 to A s though fidig the iverse. Wht hppes? 3. Write dow the simulteous equtios i the form Ax b. Solve this by 3x 3x x 6 3 x x 3x 3 3 x 5x x (i) fidig the iverse of A d computig (ii) writig dow A d b side-by-side: x A b ; d pplyig the ecessry row opertios to reduce the left side of the mtrix to the idetity mtrix. 4. Fid the iverse of the mtrix D Athoy Ty 7-5

34 Sectio 8 A Itroductio to Determits d Crmer s Rule I this sectio, you re itroduced to formul for solvig systems of simulteous equtios, clled Crmer s Rule. We begi with solutios to systems with two equtios i two ukows, d move our wy up to the geerl equtios i ukows cse. The formul i the geerl cse is demostrted here but ot derived; the derivtio will be give lter. The key igrediet of the formul for solvig systems of equtios is the determit of mtrix. Determits of ( ) Mtrices A system of two simulteous equtios i two ukows c be writte s Ax b, where A, x x b x x b x b x x, d b b. We hve see two (itimtely relted) wys to solve system of equtios. Oe wy is by elimitio; other wy is by computig the iverse of A d the fidig x A b. The elimitio method for computig the iverse of A clerly shows the coectio betwee the two methods. I this sectio, we cosider yet other method, cetered o the cocept of determit (which s you will see lter, is lso closely coected to the iverse of A ). By solvig the system i the usul wy, we c show tht the geerl solutio to the system is x Some observtios: b b b b d x - The deomitors of both x d x re the sme, d re mde up from oly the elemets of the coefficiet mtrix A. We cll the expressio i the deomitor the determit of A, deoted by A Determits re lso sometimes deoted s. det()a, or. Athoy Ty 8-

35 - If the deomitor is zero, the system will ot hve uique solutio, sice we cote divide by zero (the system will hve either o solutio, or ifiite umber of solutios). This expressio therefore determies whether or ot the system of equtios hs uique solutio; the system will hve uique solutio oly if A 0. - Observe tht the umertor of x c be expressed s the determit of the b mtrix b, which is just the mtrix A with its first colum replced by b b b. Likewise, the umertor of x is the determit of the mtrix b, b which is the mtrix A with its secod colum replced by. b I other words, the solutios c be writte i terms of determits s x b b d x b b Why is this importt? Becuse this ptter exteds to lrger systems of equtios (it is clled Crmer s Rule ). Furthermore, by usig the properties of Determits, we c (i) fid wys of helpig us compute solutios of lrger systems more esily, d (ii) we c ofte sy lot regrdig the chrcteristics of solutios of systems of equtios, without ctully solvig them. More o ll this lter. For ow, memorize the formul for the determit of ( ) mtrix. Whe will the determit of the coefficiet mtrix be zero? There re the obvious cses where ll the elemets oe or more row or oe or more colum re zero: 0 0 0, 0 0 0, It might be good exercise for you to write out the systems of equtios tht correspod to these coefficiet mtrices. Athoy Ty 8-

36 A less obvious cse is whe oe row is multiple of the other (or is the sme s the other): c c c c 0 ; Tke for exmple the simulteous equtios systems x 4x 3x 6x.5 x 4x 3x 6x The coefficiets i the secod equtio i both systems of equtios re.5 times of the first equtio. You c quickly clculte the determit of the coefficiet mtrix to be zero. For the system of equtios o the left, the costt o the right hd side of the equlity for the secod equtio is lso.5 times tht of the first. I this cse, the two equtios re the essetilly the sme: dividig the secod equtio throughout by.5 gives the first. There is effectively oly oe equtio here, i two ukows, so there re ifiitely my solutios. The system o the right, o the other hd, obviously hs o solutio. Oe specil cse should be cosidered seprtely. Cosider the system x x x x Such system is clled homogeous system. If the determit of the coefficiet mtrix A is o-zero, the there is uique solutio x 0 d x 0. This is sometimes referred to s the trivil solutio (these re just two lies tht itersect t the origi). The questio the is: whe will this system of equtios hve o-trivil solutio? It should be strightforwrd to see tht o-trivil solutios will exist oly if the determit of the coefficiet mtrix is zero. (Crmer s rule will ot give you the solutios i this cse, however you get 0/0 result). 0 0 Athoy Ty 8-3

37 Determits of (3 3) Mtrices The geerl 3 equtio 3 ukow system x x x b 3 3 x x x b 3 3 x x x b tkes bit more work to solve, but with little bit of ptiece you c show tht the solutio is x x x 3 b 33 3b3 3b 3 3b3 b 33 b b 33 b 33 3b 3 3b 3 3b3 b b b b b b b Obviously you would t wt to memorize this, t lest ot i this form. However, observe gi tht the deomitors of ll three re the sme, d composed oly of the coefficiets (the s). Agi, the system will hve uique solutio oly if the expressio i the deomitors does ot equl zero; if it equls zero, the the system will either hve ifiitely my solutios, or oe. The expressio i the deomitor gi determies whether or ot the system will hve uique solutio, so we will collect the coefficiets ito the mtrix d defie the determit of this (33) A mtrix to be A This formul is lso ot esily memorized. There is shortcut tht is sometimes useful: extedig the mtrix to iclude the first two colums Athoy Ty 8-4

38 The formul is the esily remembered i terms of the digols where the terms uder the solid rrows re dded, wheres the terms uder the dshed rrows re subtrct. Next, observe tht the umertors of x, x, d x 3 c the be writte s b 3 b 3 b respectively, i.e. we c write, b 3 b 3 b , d b b b x b 3 b 3 b3 3 33, x b 3 b 3 3 b3 33, x 3 b b 3 3 b We will sve ourselves some tedium by usig Ai () b b the i th colum replced by b b, thus writig b3 to represet the mtrix A with x A () b A, () x A b A, d 3() x3 A b A. Exmple Use Crmer s Rule to solve y z 7 x y 3z 3x 3y z 0 The coefficiet mtrix is 0 A Usig the shortcut rule for computig determits, we hve Athoy Ty 8-5

39 which gives A 0( 8)( 3) Also, A() b 3, d the determit of this mtrix is which gives () A ( b 4) 0( 6) 0( 63) x 8, so the solutio for x is Similrly, the mtrices A () b d A 3 () b re A() b 3 d A3 0 7 () b with determits () A b 7 d () A 3 b 54, so the full solutio is 35 x, 8 7 y, d z. 8 4 The visul trick for computig (3 3) mtrices does ot exted to lrger squre mtrices. I the ext sectio, we will look t differet formul for the determit, which will pply geerlly. Athoy Ty 8-6

40 Exercises. Fid the determits of the followig mtrices (i) (ii) 3 4 (iii) 4 (iv) 0 0 (v) 0 0 (vi) (vii) (viii) 3 (ix) 0 0 (x) 7 4. Solve the followig systems of equtios usig Crmer s Rule (i) x x 4 x x (ii) 3x 5x 6 6x 0x (iii) 3x 5x 6 6x 0x 0 (iv) y x 4 y x (v) x 3x 0 x x 0 (vi) 3x 5x 0 6x 0x 0 For the systems tht do ot hve uique solutio, fid out whether it hs zero or ifiitely my solutios. 3. Solve, usig Crmer s rule, the followig system of equtios for C d Y (tke everythig else s fixed) C by, 0 b Y C G. 4. Suppose we hve the followig demd d supply equtio d Q P, 0 0 d s Q P R, 0, 0 s 0 where Q d Q re the qutities demded (by cosumers) d supplied (by firms). Suppose tht i equilibrium, P is the mrket price of the good, d R is rifll. I equilibrium, we hve d s Q Q () Q. Usig Crmer s Rule, solve this system of equtios for the equilibrium price P d qutity Q, tretig everythig else s fixed. Wht hppes to equilibrium price d qutity whe rifll R icreses? How would your swer compre with the cse whe demd is completely ielstic (i.e. 0 )? Athoy Ty 8-7

41 5. Fid the determits of the followig mtrices (i) (ii) Use Crmer s Rule to solve 4x z 4 9x y 3z 3 7x y 7 Let A 3 4 d b b B b3 b. Prove tht AB A B 4 This is very useful result, d holds for the geerl cse, ot oly for ( ) cse: if A d B re () mtrices, the AB A B. Note tht it is essetil, however, for both A d B to be squre mtrices of the size dimesio (why?) 8. Fid the determits of (i) 0 b d (ii) c 0 d (iii) 0 0 b 9. Fid the determits of b c (i) 0 e f 0 0 i (ii) 0 0 d e 0 g h i 0. Let 3 A () Show tht the third equtio c be writte s lier combitio of the first two rows (i.e., you c fid c d c such tht (b) Show tht det A 0. c 3 c Athoy Ty 8-8

42 Sectio 9 The Lplce Expsio I the lst sectio, we defied the determit of (3 3) mtrix A to be A I this sectio, we itroduce geerl formul for computig determits. Rewritig A ( ) ( ) ( ) ( ) ( ) ( ) ote tht the terms outside the brckets re the terms log the first row of the mtrix The term i brckets ssocited with is the determit of the ( ) mtrix fter deletig the st row d st colum of A : ( ) The term i brckets ssocited with is the determit of the ( ) mtrix fter deletig the st row d d colum of A : ( ) The term i brckets ssocited with 3 is the determit of the ( ) mtrix fter deletig the st row d 3 d colum of A : ( ) Athoy Ty 9-

43 There is the mtter of the mius sig i frot of. This c be chieved by multiplyig ito ech term ( ) i j. Therefore, we c write ( ) ( ) ( ) (,)th Mior of A (,)th Mior of A (,3)th Mior of A (,)th Cofctor of A (,)th Cofctor of A (,3)th Cofctor of A Some mes hve bee itroduced i the formul bove: the determit of the mtrix fter removig the i th row d j th colum is clled the ( i, j )th mior of A ; icludig the sig ( ) i j gives us the ( i, j )th cofctor of A. Exmple The determit of the mtrix 0 A is A (0)( ) ()( ) ( )( ) ; The cofctor expsio for computig determits is ot uique. For istce, we could hve writte the origil formul s A ( ) ( ) ( ) which c be writte s Athoy Ty 9-

44 ( ) 3 ( ) 3 3 ( ) (,)th Mior of A (,)th Mior of A (,3)th Mior of A (,)th Cofctor of A (,)th Cofctor of A (,3)th Cofctor of A Altertively, we could hve expded log colum: A i which cse, we hve ( ) ( ) ( ) ( ) ( ) ( ) (,)th Mior of A (,)th Mior of A (3,)th Mior of A (,)th Cofctor of A (,)th Cofctor of A (3,)th Cofctor of A Note tht we chieved the correct sigs by multiplyig the ( i, j )th mior with ( ) i j. Expdig log y row or colum would i fct give us the sme expressio; we c write or 3 ( ) i A j M for y row i j 3 i ij ( ) i A j M for y colum j. ij ij ij where M ij is the ( i, j )th mior of A. This formul is kow s the Lplce Expsio. Athoy Ty 9-3

45 The fct tht you c expd log y row or colum c simplify computtios substtilly if there is row or colum with my zeros. Exmple Compute the determit of the mtrix 0 A usig the Lplce Expsio (i) expdig log the first row, (ii) expdig dow the secod colum ()( ) (0)( ) ()( ) (i) 3 0 (ii) (0)( ) (0)( ) (4)( ) 76 3 () Determits Crmer s Rule d the Lplce Expsio exted to lrger systems. The determit for geerl ( ) mtrix is or A ( ) i j j ij A M for y row i ( ) i j i ij A M for y colum j where the (, )th i j mior M ij of mtrix A is the determit of the ()() mtrix tht remis fter removig the ith row d jth colum of A. ij ij Athoy Ty 9-4

46 Exmple The determit of D is D 3( ) 3 0( ) ( ) 0 ( )( ) where we hve expded dow the third colum. The Lplce expsio icludes the ( ) cse. Defie the determit of sigle umber s. Note tht i this cotext the symbol does ot refer to bsolute vlues, e.g. =, ot. The tkig, sy, first row expsio, we hve A j j( ) M j ( ) M ( ) M j. Crmer s Rule lso exteds to geerl ( ) systems of equtios. The solutio to x x... x b x x... x b... x x... x b is i ( ) xi A b A, i,...,, where b b A, b b d Ai ( b) is the mtrix A with its i th colum replced by b. Athoy Ty 9-5

47 Exercises. Fid the determits of the followig mtrices usig the Lplce expsio. (i) (ii) (iii) (vi) (v) (vi) (vii) (viii) (ix) Use Crmer s Rule to solve 4x z 4 9x y 3z 3 7x y 3. Solve the followig system of equtios x x 3x 4x 3 4 x x 3 4 x x 3x 4 x x 3x Fid the determits usig the Lplce expsio (i) (ii) (iii) (iv) (v) (vi) Athoy Ty 9-6

48 5. Let A be rbitrry ( ) mtrix. Show tht if we multiply every elemet of sigle row or colum by c, the the determit of the ew mtrix is c A. Wht is the determit of ca? 6. Let E be the mtrix obtied by switchig the st d lst rows of ( ) idetity mtrix, i.e E () Show usig the Lplce expsio tht det E ; (b) Does your swer deped o whether is eve or odd? (c) Use your result to prove tht the determit of the mtrix formed by switchig y two rows of the idetity mtrix is. 7. () Let 0 0 E d suppose A is some (3 ) mtrix. Describe the rows of the product EA i terms of the rows of A ; Wht is the determit of the mtrix E? (b) Let E be mtrix tht crries out the elemetry row opertio of subtrctig multiple of oe row from other row. Wht is the determit of E? Athoy Ty 9-7

49 Sectio 0 Properties of Determits A very importt result, stted here without proof, is tht AB A B if A d B re squre mtrices tht c be multiplied together. We c use this result to relte elemetry row opertios to the determits of squre mtrices, geertig severl importt properties of determits. This i tur provides wy to simplify the computtio of determits (icludig determits of lrger mtrices). () If two rows of A re iterchged, the sig of the determit chges. () If ll the elemets of sigle row or colum of A re multiplied by, the the determit of the ew mtrix is A. (3) If multiple of row (colum) is dded to other row (colum), the determit remis uchged. We illustrte these usig exmples. It should be esy for you to geerlize from these exmples. Throughout, we use A s exmple. You c verify tht the determit of this mtrix is 4. Recll tht switchig two rows of the mtrix: e.g. switchig rows d : is equivlet to premultiplyig A with tht mtrix You c verify this: E ()() A E ()() Athoy Ty 0-

50 The mtrix E()() is obtied by switchig the st d d rows of the (3 3) idetity mtrix. Usig the Lplce expsio, you c esily show tht E ()() This is true i geerl: the determit of the idetity mtrix is oe; whe two rows of the idetity mtrix re switched, the determit of the ew mtrix becomes. This mes tht switchig two rows of mtrix results i switch i the sig of its determit. I our exmple: 3 E A E A A ()()()() 3 3 ( )( E4) A4 E A 5 ()()()() I geerl, switchig two rows of mtrix switches the sig of its determit. The secod type of row opertor is multiplyig row of mtrix by some umber: e.g. multiply row of A by /3: This is equivlet to premultiplyig the mtrix with E () You c esily verify tht the determit of this mtrix is 3. Therefore ( 4) E A8 E A 5 ()() 3 Multiplyig oe row of mtrix by multiplies the determit by. Athoy Ty 0-

51 Filly, the third type of elemetry row opertor is to dd/subtrct costt times oe row to other row: e.g. subtrct hlf of row of A from row 3: This is equivlet to premultiplyig the mtrix with E (3)()(3) obtied by subtrctig hlf of the secod row of the idetity mtrix from row 3 of the idetity mtrix. It is obvious tht the determit of this mtrix is oe. Therefore 3 3 3() E 4 A E A A (3)()(3)(3)()(3) Addig/subtrctig costt times oe row of mtrix to other row of the mtrix does ot chge its determit. Sice the determit of trigulr mtrix is simply the product of the elemets i the digol, these results imply tht to compute determit, we c reduce mtrix to trigulr mtrix, compute the determit of tht, d reverse the effects of the row opertios used. Row Opertio gives Effect o det switch rows d 3, [ ( ) ] row mius times row, [o chge] 3 3 row 3 mius 3 times row, [o chge] Athoy Ty 0-3

52 Row Opertio gives Effect o det. row mius row 3, row 3 plus 4 times row, [o chge] [o chge] Determit of this lst trigulr mtrix is ( 3 8 ) = 4. Lookig bck t the row opertios used, we fid we switch the sig oce. Therefore, the determit of the origil mtrix is Exercises The followig mtrices re derived from the mtrix usig elemetry row opertios. Fid their determits (i) 3 (ii) 4 6 (iii) It is true geerlly tht A A. Verify this for ( ) d (3 3) mtrices. 3. Fid the determit of the mtrix D by first reducig to trigulr mtrix. Athoy Ty 0-4

53 Sectio Rk d Lier Depedece I erly sectio, we lert tht the umber of solutios i system of lier equtios depeds o the umber of idepedet equtios i the system reltive to the umber of ukows. From Crmer s Rule, we lert tht system of -lier equtios i -ukows will hve uique solutio oly whe the determit of the coefficiet mtrix is ot zero. Whe will the determit of mtrix be zero? This hs lot to do with the lier depedece i the coefficiet mtrix. The presece of y zero row or colum will produce zero determit (this is esy to verify: just use the Lplce expsio log tht zero row or colum). I the ( ) cse we obtied zero determit whe oe row ws multiple of the other (or is the sme s the other), d this remis true of the (33) cse. A more subtle cse is whe oe row is lier combitio of the other two (i.e., we c write oe row s the sum of some multiple of secod row plus some multiple of the third). For istce, tke the mtrix 3 Α which you c verify hs zero determit. Here the third row is equl to the first row plus hlf of the secod row: (Altertively, we c sy tht the first row is the third row mius hlf of the secod or tht the secod row is two times the third row mius two times the first.) We sy tht the three row vectors tht mke up the mtrix re lierly depedet. This lier depedece betwee the rows leds to zero determit becuse it mes tht row opertios c be pplied to ultimtely crete zero row. A (3 3) mtrix will hve o-zero determit oly if it hs three lierly idepedet rows (or colums). More geerlly, thik of (3 3) squre mtrix s stck of three row vectors A Athoy Ty -

54 A We sy tht three (colum or row) vectors,, d 3 there re costts c, c, d c 3, ot ll zero, such tht c c c re lierly depedet if If the oly cse where this is true is c c c3 0, the the vectors re sid to be lierly idepedet. For ow, defie the rk of the mtrix is the umber of lierly idepedet row vectors tht it cotis. Exmples. Let A. Let,, d The these vectors re lierly depedet, sice The secod row is just twice tht of the first row (or the first is hlf of the secod). Removig oe of the depedet vectors, sy 4 6, the remiig vectors d 3 re lierly idepedet: the oly costts c d c such tht c 3 c re c c 0. The mtrix hs two lier idepedet vectors, d is sid to be of rk. If we imgie these three vectors i the usul represettio s rrows i threedimesiol co-ordite spce, we will see tht the vector is just extesio of. Combitios of the three vectors re effectively combitios of oly two vectors. c c c c c ( c) c c c Differet combitios of the three vectors will therefore result i ew vectors tht ll lie o ple ( two-dimesiol spce), d cot sp or cover the etire 3-d spce.. Let A. Let,, d The these vectors re lierly depedet, sice The secod row depeds o the other two i tht it is the sum of d 3. Athoy Ty -

55 (Equivletly, 3 depeds o d becuse it is ). Removig oe of these depedet vectors (y oe of them) leves two idepedet vectors. For istce, removig 3, the the oly costts c d c such tht c c c c re c c 0. The mtrix hs two lier idepedet vectors, d is sid to be of rk. If we imgie these three vectors geometriclly i three-dimesiol co-ordite spce, we will see tht lthough o oe vector is extesio of other, ll three vectors oetheless lie i sigle ple, so gi combitios of the three vectors will crete ew vectors tht lso lie o tht ple. The three vectors cot sp, or cover, the etire 3-d spce. 3. Let A. Let,, d The these vectors re lierly depedet, sice I 3 this cse, there is oly oe lierly idepedet vector. If we remove 3, we fid tht the remiig two re still lierly depedet: is twice tht of. The rk of this mtrix is oe: rk() A. Geometriclly, ll three vectors lie o sigle lie. The vector is twice tht of, d 3 is three times tht of. Combitios of these vectors therefore will oly crete ew vectors tht re lso extesios of. c c c c c c 3( c 3) c c Let A. Let,, d The these vectors re lierly depedet, sice c for y c. Removig vector leves two lierly idepedet vectors. The rk of this mtrix is two. Athoy Ty -3

56 5. Let A. Let,, d The ll three row vectors re lierly idepedet; it is impossible to fid c, c d c 3, ot ll zero, such tht c c c This mtrix is of full rk. A squre mtrix will hve o-zero determit if d oly if ( iff ) it hs full rk. There re my wys to determie the rk of mtrix. Oe wy is to do Gussi Elimtio ( the row opertios) d see how my o-zero pivots you get (see Sectio for pivots ). Other wys will be discussed i lter sectios, or i more dvced clsses. For the time beig we will proceed by observtio. We coclude this sectio with severl remrks regrdig rk. Erlier, we viewed mtrix s stck of row vectors. The rk cocept tht we discussed could be clled row rk. But we c lso view mtrix s the coctetio (joiig together) of three colum vectors A. Everythig tht we sid here follows for colum vectors s for row vectors, d i fct mtrix will hve the sme umber of lierly idepedet colum vectors s there re lierly idepedet row vectors: if A hs two lierly idepedet row vectors, it will lso hve two lierly idepedet colum vectors; if it hs oly oe lierly idepedet row vector, the it will oe hve oe lierly idepedet colum vector, d so o. A mtrix s row rk is the sme s its colum rk.. We c lso spek of the rk of o-squre mtrices. Tke for exmple 4 6 A 3 This is i fct the lst two rows of erlier exmple we were lookig t. Lookig t the rows, we see tht these two rows re lierly idepedet (verify!) The row - rk is two. Wht if we view this mtrix s coctetio of three two-dimesiol vectors? Note tht three two-dimesiol vectors c oly sp two-dimesiol Athoy Ty -4