Chapter System of Equations


 Camron Lamb
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1 hpter System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vicevers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee costet d icostet system of lier equtios, d 4. ler tht system of lier equtios c hve uique solutio, o solutio or ifiite solutios. Mtrix lgebr used for solvig systems of equtios. you illustrte th cocept? Mtrix lgebr used to solve system of simulteous lier equtios. I fct, for my mthemticl procedures such s the solutio to set of olier equtios, iterpoltio, itegrtio, d differetil equtios, the solutios reduce to set of simulteous lier equtios. Let us illustrte with exmple for iterpoltio. Exmple The upwrd velocity of rocket give t three differet times o the followig tble. Tble 5.. Velocity vs. time dt for rocket Time, t Velocity, v (s) (m/s) The velocity dt pproximted by polyomil s v ( t) t + bt + c, 5 t. Set up the equtios i mtrix form to fid the coefficiets, b, c of the velocity profile. Solutio The polyomil goig through three dt poits (, v ), ( t, v ), d ( t v ) tble 5.. t, v t, v t where from 3,
2 hpter t, v v t t + bt + c psses through the three dt poits gives 3 Requirig tht ( ) v ( t ) v t + bt + c v ( t ) v t + bt + c v ( t3 ) v3 t3 + bt3 + c Substitutig the dt (, v ), ( t, v ), d ( t, v ) ( 5 ) + b( 5) + c ( 8 ) + b( 8) + c 77. ( ) + b( ) + c 79. t gives 3 3 or 5 + 5b + c b + c b + c 79. Th set of equtios c be rewritte i the mtrix form s 5 + 5b + c b + c b + c 79. The bove equtio c be writte s lier combitio s follows b 8 c d further usig mtrix multiplictio gives b c 79. The bove illustrtio of why mtrix lgebr eeded. The complete solutio to the set of equtios give lter i th chpter. A geerl set of m lier equtios d ukows, x + x + + x c x + x + + x c. m x + m x m x cm c be rewritte i the mtrix form s
3 System of Equtios x c.. x c m m.. m x cm A, [ X ], d [ ], the system of equtio A clled the coefficiet mtrix, [ ] clled the right hd side X clled the solutio vector. A X systems of equtios re writte i the ugmeted form. Tht Deotig the mtrices by [ ] [ A ][ X ] [ ], where [ ] vector d [ ] Sometimes [ ][ ] [ ]... c... c [ A ] m m... m c A system of equtios c be costet or icostet. Wht does tht me? A system of equtios [ ][ X ] [ ] A costet if there solutio, d it icostet if there o solutio. However, costet system of equtios does ot me uique solutio, tht, costet system of equtios my hve uique solutio or ifiite solutios (Figure ). [A][X] [B] ostet System Icostet System Uique Solutio Ifiite Solutios Figure 5.. ostet d icostet system of equtios flow chrt. Exmple Give exmples of costet d icostet system of equtios. Solutio ) The system of equtios
4 hpter x 6 3y 4 costet system of equtios s it hs uique solutio, tht, x. y b) The system of equtios 4 x 6 y 3 lso costet system of equtios but it hs ifiite solutios s give s follows. Expdig the bove set of equtios, x + 4y 6 x + y 3 you c see tht they re the sme equtio. Hece, y combitio of ( y) x + 4y 6 solutio. For exmple (, y) (, ) ( x, y) (0.5,.5), ( x, y) (0,.5), d so o. c) The system of equtios 4 x 6 y 4 icostet s o solutio exts. x, tht stfies x solutio. Other solutios iclude How c oe dtiguh betwee costet d icostet system of equtios? A system of equtios [ ][ X ] [ ] ugmeted mtrix [ A ] A system of equtios [ ][ X ] [ ] the ugmeted mtrix [ A ]. A costet if the rk of A equl to the rk of the A icostet if the rk of A less th the rk of But, wht do you me by rk of mtrix? The rk of mtrix defied s the order of the lrgest squre submtrix whose determit ot zero. Exmple 3 Wht the rk of 3 [ A ] 0 5? 3 Solutio The lrgest squre submtrix possible of order 3 d tht [A] itself. Sice det( A ) 3 0, the rk of [ A ] 3.
5 System of Equtios Exmple 4 Wht the rk of 3 [ A ] 0 5? 5 7 Solutio The lrgest squre submtrix of [A] of order 3 d tht [A] itself. Sice det( A ) 0, the rk of [A] less th 3. The ext lrgest squre submtrix would be mtrix. Oe of the squre submtrices of [A] 3 [ B ] 0 d det( B ) 0. Hece the rk of [A]. There o eed to look t other submtrices to estblh tht the rk of [A]. Exmple 5 How do I ow use the cocept of rk to fid if 5 5 x x x costet or icostet system of equtios? Solutio The coefficiet mtrix 5 5 [ A ] d the right hd side vector 06.8 [ ] The ugmeted mtrix [ B ] Sice there re o squre submtrices of order 4 s [B] 3 4 mtrix, the rk of [B] t most 3. So let us look t the squre submtrices of [B] of order 3; if y of these squre
6 hpter submtrices hve determit ot equl to zero, the the rk 3. For exmple, submtrix of the ugmeted mtrix [B] 5 5 [D] hs det( D ) Hece the rk of the ugmeted mtrix [B] 3. Sice [ A ] [ D], the rk of [A] 3. Sice the rk of the ugmeted mtrix [B] equls the rk of the coefficiet mtrix [A], the system of equtios costet. Exmple 6 Use the cocept of rk of mtrix to fid if 5 5 x x x costet or icostet? Solutio The coefficiet mtrix give by 5 5 [ ] A d the right hd side 06.8 [ ] The ugmeted mtrix 5 5 :06.8 [ B ] 64 8 : : 84.0 Sice there re o squre submtrices of order 4 s [B] 4 3 mtrix, the rk of the ugmeted [B] t most 3. So let us look t squre submtrices of the ugmeted mtrix [B] of order 3 d see if y of these hve determits ot equl to zero. For exmple, squre submtrix of the ugmeted mtrix [B] 5 5 [ D ]
7 System of Equtios hs det( D ) 0. Th mes, we eed to explore other squre submtrices of order 3 of the ugmeted mtrix [B] d fid their determits. Tht, [ E ] det( E ) [ F ] det( F ) [ G ] det( G ) 0 All the squre submtrices of order 3 3 of the ugmeted mtrix [B] hve zero determit. So the rk of the ugmeted mtrix [B] less th 3. Is the rk of ugmeted mtrix [B] equl to?. Oe of the submtrices of the ugmeted mtrix [B] 5 5 [ H ] 64 8 d det( H ) 0 0 So the rk of the ugmeted mtrix [B]. Now we eed to fid the rk of the coefficiet mtrix [B]. 5 5 [ A ] d det( A ) 0 So the rk of the coefficiet mtrix [A] less th 3. A squre submtrix of the coefficiet mtrix [A] 5 [ J ] 8 det( J ) 3 0 So the rk of the coefficiet mtrix [A].
8 hpter Hece, rk of the coefficiet mtrix [A] equls the rk of the ugmeted mtrix [B]. So the system of equtios [ A ][ X ] [ ] costet. Exmple 7 Use the cocept of rk to fid if 5 5 x x x costet or icostet. Solutio The ugmeted mtrix 5 5 :06.8 [ B ] 64 8 : : 80.0 Sice there re o squre submtrices of order 4 4 s the ugmeted mtrix [B] 4 3 mtrix, the rk of the ugmeted mtrix [B] t most 3. So let us look t squre submtrices of the ugmeted mtrix (B) of order 3 d see if y of the 3 3 submtrices hve determit ot equl to zero. For exmple, squre submtrix of order 3 3 of [B] 5 5 [ D ] det(d) 0 So it mes, we eed to explore other squre submtrices of the ugmeted mtrix [B] [ E ] det( E ).0 0. So the rk of the ugmeted mtrix [B] 3. The rk of the coefficiet mtrix [A] from the previous exmple. Sice the rk of the coefficiet mtrix [A] less th the rk of the ugmeted mtrix [B], the system of equtios icostet. Hece, o solutio exts for [ A ][ X ] [ ]. If solutio exts, how do we kow whether it uique? I system of equtios [ A ][ X ] [ ] tht costet, the rk of the coefficiet mtrix [A] the sme s the ugmeted mtrix [ A ]. If i dditio, the rk of the coefficiet mtrix [A] sme s the umber of ukows, the the solutio uique; if the rk of the coefficiet mtrix [A] less th the umber of ukows, the ifiite solutios ext.
9 System of Equtios [A] [X] [B] ostet System if rk (A) rk (A.B) Icostet System if rk (A) < rk (A.B) Uique solutio if rk (A) umber of ukows Ifiite solutios if rk (A) < umber of ukows Figure 5.. Flow chrt of coditios for costet d icostet system of equtios. Exmple 8 We foud tht the followig system of equtios 5 5 x x x costet system of equtios. Does the system of equtios hve uique solutio or does it hve ifiite solutios? Solutio The coefficiet mtrix 5 5 [ A ] d the right hd side 06.8 [ ] While fidig out whether the bove equtios were costet i erlier exmple, we foud tht the rk of the coefficiet mtrix (A) equls rk of ugmeted mtrix [ A ] equls 3. The solutio uique s the umber of ukows 3 rk of (A). Exmple 9 We foud tht the followig system of equtios 5 5 x x x costet system of equtios. Is the solutio uique or does it hve ifiite solutios.
10 hpter Solutio While fidig out whether the bove equtios were costet, we foud tht the rk of the coefficiet mtrix [A] equls the rk of ugmeted mtrix ( A ) equls Sice the rk of [ A ] < umber of ukows 3, ifiite solutios ext. If we hve more equtios th ukows i [A] [X] [], does it me the system icostet? No, it depeds o the rk of the ugmeted mtrix [ ] A d the rk of [A]. ) For exmple x x x costet, sice rk of ugmeted mtrix 3 rk of coefficiet mtrix 3 Now sice the rk of (A) 3 umber of ukows, the solutio ot oly costet but lso uique. b) For exmple x x x icostet, sice rk of ugmeted mtrix 4 rk of coefficiet mtrix 3 c) For exmple x x x costet, sice rk of ugmeted mtrix rk of coefficiet mtrix But sice the rk of [A] < the umber of ukows 3, ifiite solutios ext. ostet systems of equtios c oly hve uique solutio or ifiite solutios. system of equtios hve more th oe but ot ifiite umber of solutios? No, you c oly hve either uique solutio or ifiite solutios. Let us suppose [ A ][ X ] [ ] hs two solutios [Y ] d [Z] so tht
11 System of Equtios [ A ][ Y ] [ ] [ A ][ Z] [ ] If r costt, the from the two equtios r [ A][ Y ] r[ ] ( r)[ A][ Z ] ( r)[ ] Addig the bove two equtios gives r[ A][ Y ] + ( r)[ A][ Z ] r[ ] + ( r)[ ] [ A ]( r[ Y ] + ( r)[ Z ]) [ ] Hece r[ Y ] + ( r)[ Z ] solutio to [ A ][ X ] [ ] Sice r y sclr, there re ifiite solutios for [ A ][ X ] [ ] of the form r Y + r Z [ ] ( )[ ] you divide two mtrices? If [ A ][ B] [ ] defied, it might seem ituitive tht [ ] [ B] [ A ], but mtrix divio ot defied like tht. However iverse of mtrix c be defied for certi types of squre mtrices. The iverse of squre mtrix [A], if extig, deoted by [ A ] such tht [ A][ A] [ I] [ A] [ A] Where [I] the idetity mtrix. I other words, let [A] be squre mtrix. If [B] other squre mtrix of the sme size such tht [ B ][ A] [ I], the [B] the iverse of [A]. [A] the clled to be ivertible or osigulr. If [ A ] does ot ext, [A] clled oivertible or sigulr. If [A] d [B] re two mtrices such tht [ B ][ A] [ I], the these sttemets re lso true [B] the iverse of [A] [A] the iverse of [B] [A] d [B] re both ivertible [A] [B][I]. [A] d [B] re both osigulr ll colums of [A] d [B]re lierly idepedet ll rows of [A] d [B] re lierly idepedet. Exmple 0 Determie if 3 [B] 5 the iverse of 3
12 hpter [A] 5 Solutio [ B][ A] [I] Sice [ B ][ A] [ I], [B] the iverse of [A] d [A] the iverse of [B]. But, we c lso show tht 3 3 [ A][ B] [I] to show tht [A] the iverse of [B]. I use the cocept of the iverse of mtrix to fid the solutio of set of equtios [A] [X] []? Yes, if the umber of equtios the sme s the umber of ukows, the coefficiet mtrix [A] squre mtrix. Give [ A ][ X ] [ ] The, if [ A ] exts, multiplyig both sides by [ A ]. [ A] [ A][ X ] [ A] [ ] [ I][ X ] [ A] [ ] [ X ] [ A] [ ] Th implies tht if we re ble to fid [ A ], the solutio vector of [ A ][ X ] [ ] simply multiplictio of [ A ] d the right hd side vector, []. How do I fid the iverse of mtrix? If [A] mtrix, the [ iverse of mtrix [ A ][ A] [ I] Deotig A ] mtrix d ccordig to the defiitio of
13 System of Equtios A ] [ ] [ A ] [I Usig the defiitio of mtrix multiplictio, the first colum of the ] [ A mtrix c the be foud by solvig 0 0 Similrly, oe c fid the other colums of the ] [ A mtrix by chgig the right hd side ccordigly. Exmple The upwrd velocity of the rocket give by Tble 5.. Velocity vs time dt for rocket Time, t (s) Velocity, v (m/s) I erlier exmple, we wted to pproximte the velocity profile by ( ) 5, + + t c bt t t v We foud tht the coefficiets c b d,, i ( ) t v re give by
14 hpter b c 79. First, fid the iverse of 5 5 [ A ] d the use the defiitio of iverse to fid the coefficiets, b,d c. Solutio If [ A ] 3 3 the iverse of [A], the gives three sets of equtios Solvig the bove three sets of equtios seprtely gives
15 System of Equtios Hece Now where [ A ] [ A ][ X ] [ ] [ X ] [ ] b c Usig the defiitio of [ A ], [ A] [ A][ X ] [ A] [ ] [ X] [ A ] [ ] Hece So b 9.69 c.086 v ( t) 0.905t t +.086, 5 t Is there other wy to fid the iverse of mtrix? For fidig the iverse of smll mtrices, the iverse of ivertible mtrix c be foud by [ A] dj( A) det( A) where
16 hpter where dj ( A) ij re the cofctors of ij T. The mtrix itself clled the mtrix of cofctors from [A]. ofctors re defied i hpter 4. Exmple Fid the iverse of 5 5 [ A ] Solutio From Exmple 4.6 i hpter 04.06, we foud det( A ) 84 Next we eed to fid the djoit of [A]. The cofctors of A re foud s follows. The mior of etry 5 5 M The cofctors of etry + ( ) M M 4 The mior of etry 5 5 M
17 System of Equtios The cofctor of etry + ( ) M M ( 80) 80 Similrly Hece the mtrix of cofctors of [A] [ ] The djoit of mtrix [A] Hece ( A) [ ] T dj [ A] det ( A) dj T [], ( A)
18 hpter If the iverse of squre mtrix [A] exts, it uique? Yes, the iverse of squre mtrix uique, if it exts. The proof s follows. Assume tht the iverse of [A] [B] d if th iverse ot uique, the let other iverse of [A] ext clled []. If [B] the iverse of [A], the [ B ][ A] [ I] Multiply both sides by [], [ B ][ A][ ] [ I][ ] [ B ][ A][ ] [ ] Sice [] iverse of [A], [ A ][ ] [ I] Multiply both sides by [B], [ B ][ I] [ ] [ B ] [ ] Th shows tht [B] d [] re the sme. So the iverse of [A] uique. Key Terms: ostet system Icostet system Ifiite solutios Uique solutio Rk Iverse
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