Chapter 6: Determiats ad the Iverse Matrix SECTION E pplicatios of Determiat By the ed of this sectio you will e ale to apply Cramer s rule to solve liear equatios ermie the umer of solutios of a give liear system I this sectio we examie a alterative method for solvig liear systems of equatios. Cramer s rule allows us to fid the solutio to a system of equatios without havig to fid the iverse, as it is ased etirely o ermiats. E Cramer s Rule Gariel Cramer was or i 704 i Geeva, Switzerlad ad y the age of 8 he had received a doctorate o the theory of soud. I 74 he attaied the chair i mathematics at cademie de Clavi i Geeva ad taught geometry ad mechaics. Cramer is popularly kow i mathematics circles for his cotriutio to liear algera, appropriately called Cramer s Rule which was descried i his famous ook; Itroductio to the alysis of lgeraic Curves. This rule was kow to other mathematicias of that era ut his superior otatio is the reaso it is credited to him. Figure Gariel Cramer 704 to 75 Cramer worked very hard throughout his life, writig ooks i his spare time ad carryig out other editorial work. However, a fall from carriage comied with may years of reletless study cotriuted to his death at the age of 47. I this sectio we first state Cramer s rule ad the show how this rule ca e used to solve liear systems of equatios. Rememer a liear system of equatios ca e writte i matrix form as x, where is a m y matrix ad x ad are colum vectors, that is; a y matrix. For example we ca write the followig liear system of equatios: x 3y 7z 4 3x 5y z 6 7x 6y z 9 i matrix form as x where 3 7 x 4 3 5, x y ad = 6 7 6 z 9 How ca we solve this liear system of equatios? I previous chapters we discussed various techiques to solve liear systems such as; Gaussia elimiatio, reduced row echelo form (rref), the iverse matrix method, LU factorizatio etc. Cramer s rule gives a formula for solvig small y liear systems y calculatig a series of ermiats. This ofte leads to some much simpler arithmetic. O the dowside, it ecomes iefficiet for large systems ecause it ivolves evaluatig umerous ermiats, ad there is o easy way to work out the ermiat of a large matrix.
Chapter 6: Determiats ad the Iverse Matrix dditioally, we ca oly apply Cramer s rule to a y liear system ecause we caot evaluate the ermiat of a o-square matrix. Before we state Cramer s rule we eed to itroduce some ew otatio. Let e a y matrix ad e a y colum vector. The otatio replaces the kth colum of matrix y the colum vector. This meas that we have: a a a a k a a Therefore What does k meas the first colum i matrix is replaced y the s. mea? The secod colum i matrix is replaced y the s. What does The fifth colum i matrix is replaced y the s. Cramer s Rule (6.3). Let e a y matrix with the etries omiated y a ij 5 mea? ad e a y colum vector. The system of liear equatios x has the uique solutio: 3 x, x, x3, ad x 0 [ot equal to zero]. How do we prove this result? We have symol i the statemet which meas we eed to prove it oth ways, ad. First we prove from right to left. How do we prove this part? We assume 0 ad from this we deduce the aove equatios for the ukows x, x,, x. We are give the liear system 0 which meas that is ivertile ad y the followig result of chapter we have the solutio x is give y: (.36) x What is iverse matrix equal to? This was defied i Propositio (6.3): Propositio (6.3). If 0 the x ad We use these Propositios (.36) ad (6.3) to prove Cramer s rule. I order to use Propositio (6.3) we eed to kow what is meat y cofactor ad adjoit which were defied earlier: Defiitio (6.4). The cofactor is the mior of etry a ij. Cij of the etry ij adj a is defied as C Defiitio (6.9). The adjoit is the cofactor matrix trasposed; T. i j ij M ij where M ij adj kth Colum C. We are goig to use these defiitios ad propositios to prove Cramer s rule. If you are ot cofidet i applyig these the you will eed to go ack ad see exactly what they mea efore emarkig o the proof of this result.
Chapter 6: Determiats ad the Iverse Matrix 3 Proof.. We assume that 0 ad prove the result for x (x oe) ad the geeralize to the remaiig ukows. From the give liear system x, we have y the iverse matrix method, Propositio (.36), x where What does adj adj adj represet? [This was defied i Propositio (6.3)] is the adjoit matrix ad y (6.9) this is the cofactor matrix trasposed, that is C C C C C C adj where C s are cofactors C C C By Defiitio (6.4) the cofactor of a etry is the place sig times the ermiat of the remaiig matrix after deletig the row ad colum cotaiig that etry. The cofactors C s are give y: a a a a a a a a C ( ) a a a a a a a a a a a a a a3 a3 C ( ) a a a a a C a a a a a a a a a( ) a a a a By applyig Propositio (6.3) x adj we have x C C C x C C C x C C C ( ) C C C Multiplyig out the C matrix C C C ad the vector with row y colum multiplicatio C C C We show the result for x ad the proofs of x, x3, ad x are similar.
Chapter 6: Determiats ad the Iverse Matrix 4 Equatig x o the Left with the first etry o the Right i the aove gives : x C C C Sustitutig the ermiat give aove of the cofactor matrices What do we eed to prove? x a a a a a a a a By a a a( ) a By a3 a3 By. We have i (*) ut we eed to show the. How?. What is equal to? is the matrix ut the first colum is replaced y the colum vector, that is: expressio i the large square rackets o the Right is equal to Need to fid Well a a a a a a We ca evaluate the ermiat of this matrix y expadig alog the first colum, a a a a a a Expadig alog this a a a a colum. a3 a3 a a a a a a a a What do you otice aout this last expressio? This is idetical to the aove expressio i the square rackets i (*). By (*) we have C C C (*)
Chapter 6: Determiats ad the Iverse Matrix 5 We have x C C Similarly we ca show that for j, 3,, x j j Hece y sustitutig each j value we have 3 x, x3. If we have x The clearly, x,, ad x3 0. This completes our proof. 3 x, ad x We ca apply Cramer s rule to a particular liear system as the ext example shows. Example Solve the followig liear system y usig Cramer s rule: x y 5z 6 7x y z 7 5x 3y 3z Solutio We ca write this i matrix form x where 5 x 6 7, x y ad 7 5 3 3 z 0 ) we have pplyig Cramer s rule (provided x, y What is equal to? ad z 3 5 7 7 7 5 3 3 5 3 5 3 5 3 3 6 3 5 5 0 Hece 45 45. What else do we eed to fid? By equatios i (*) we eed to fid ermie x, y ad z respectively. What is, ad 3 equal to? (*) i order to
Chapter 6: Determiats ad the Iverse Matrix 6 6 is the matrix ut with the first colum replaced y 7 is the ermiat of this matrix: 6 5 Replacig the first 7 colum y. 3 3 7 7 6 5 3 3 3 3 6 6 3 5 45 What is. Hece equal to? is the matrix ut with the secod colum replaced y ad is the ermiat of this matrix: 6 5 Replacig the secod colum y. 7 7 5 3 7 7 7 7 6 5 3 5 3 5 6 5 5 7 35 90 What is Similarly is 3 3 is the matrix ut with the third colum replaced y ad 3 equal to? 6 3 7 7 5 3 7 7 7 7 6 3 5 5 3 Replacig the third colum y. 7 35 60 80 Sustitutig 45, 45, 90 ad 3 3 x, y ad z (*) gives 45 90 80 x, y ad z 4 45 45 45 Hece the uique solutio of the liear system is x, y ad z 4. 80 :
Chapter 6: Determiats ad the Iverse Matrix 7 E Liear Systems of Equatios Rememer liear algera is the study of liear equatios ad i this susectio we examie the relatioship etwee the solutios of a system ad the ermiat of its matrix. 0. (Theorem (6.6)). From sectio C we kow that the matrix is ivertile This meas that matrix eig ivertile is equivalet to this 0 to Theorem (.38) of chapter : 0. Hece we ca add Theorem (6.3). Let e a y matrix, the the followig 6 statemets are equivalet: (a) The matrix is ivertile (o-sigular). () The liear system x O oly has the trivial solutio x O. (c) The reduced row echelo form of the matrix is the idetity matrix I. (d) is a product of elemetary matrices. (e) x has a uique solutio. 0. (f) We have added statemet (f) to Theorem (.38) of chapter. What ca we coclude aout the liear system 0? x if Two thigs. If O [Not Zero] the x has a ifiite umer or o solutios.. If O the x O has a ifiite umer of solutios. Clearly i this case we have the trivial solutio x O which meas x 0, x 0, x3 0,, x 0 ad a ifiite umer of other solutios. Example Which of the followig liear systems have a o-trivial solutio? x y 5z 0 x y 3z 0 (a) 7x y z 0 5x 3y 3z 0 6x 7 y 8z 0 Solutio (a) Note that i this case the matrix of coefficiets is idetical to the oe i Example. Thus y the aove Example we have x O where 45 () 4x 5y 6z 0 Do we have a o-trivial solutio to the give liear system? 0 [Not Zero] therefore y the aove Propositio (6.3) part () we No ecause oly have the trivial solutio: x 0, y 0 ad z 0 () We have x O where 3 5 6 4 6 4 5 4 5 6 3 8 9 7 9 7 8 7 8 9 45 48 36 4 3 3 35 0 What ca we coclude aout the give liear system? 0 so y statemet aove we ca say that the give liear system has a Sice ifiite umer of solutios ad so has o-trivial solutios. Note that we do ot have to fid them.
Chapter 6: Determiats ad the Iverse Matrix 8 Example 3 For what values of k will the followig system have (i) a uique solutio? (ii) a ifiite umer or o solutio? x y kz 3 x ky z kx y z Solutio Writig the give liear system i matrix form we have k x 3 x where k, x y ad 3 k z The ermiat of the matrix is give y k k k k k k k 3 3 k k k k k k k k (i) Uder what coditios do we have a uique solutio? 0. Thus we have a uique solutio provided It is where k k 0 which occurs whe k 3 0 or k 0 Thus we have uique solutio provided k 0 or k. (ii) Uder what coditios do we have o or a ifiite umer of solutios? k k 0 which gives k 0 or k Thus we have o or a ifiite umer of solutios provided k 0 or k. SUMMRY Cramer s Rule (6.3). The liear system x, x 0. x has the uique solutio: 3, x3, ad x