13.3 Using Cramer s Rule to Solve Systems

Transcription

1 . Using Crmer s Rule to Solve Sstems Now tht we n solve n sstems of equtions, we wnt to lern nother tehnique for solving these sstems. This new tehnique will require us to get fmilir with severl new onepts. Let s strt with the following efinition. efinitions: Mtri A mtri is retngulr rr of numers written insie of rkets. Eh numer in mtri is lle n entr, eh horiontl set of numers is lle row n eh vertil set of numers is lle olumn. Mtries ome in wie vriet of sies. When writing the sie of mtri, we lws list the rows first. So mtri woul hve rows n olumns, for emple. Let s get hnle on these ies with the following emple. Emple: List the entries of the following mtri. Wht is the r entr of the n row n the st entr of the r olumn? Solution: Sine this mtri hs rows n olumns, it is mtri. So, with this in min, the r entr of the n row woul e, n the st entr of the r olumn woul e -. With the si ie of mtri now own, we nee to tlk out ouple of ifferent kins of mtries. Nmel, the oeffiient mtri n the ugmente mtri. efinition: Coeffiient Mtri A mtri whih is ompose of ll of the oeffiients of sstem of equtions ut oes not inlue the onstnt terms. efinition: Augmente Mtri A mtri whih is ompose of ll oeffiients n onstnts of sstem of equtions. These re firl es to fin s we see in the net emple. Emple : Write the oeffiient mtri n the ugmente mtri of the following sstems...

2 Solution:. The oeffiient mtri is the mtri tht is generte oeffiients, ut not the onstnt terms of the given sstem. The ugmente mtri is generte the oeffiients n the onstnts of the sstem. So this mens we must hve Coeffiient Mtri Augmente Mtri. Like in prt. ove, we generte the oeffiient mtri onl the oeffiients n the ugmente mtri using the entire sstem. This gives us Coeffiient Mtri Augmente Mtri There re numer of opertions n other things we n o with mtri, ut for the ske of simpliit, let s tke look t onl the opertion tht we nee for solving sstems. Tht is, the opertion of the eterminnt. efinition: eterminnt A numer ssoite with squre mtri. This numer hs numerous uses in vriet of pplitions. More importnt thn the efinition of eterminnt is how we fin eterminnt. It vries slightl epening on the sie of our mtri. We will onl look t evluting n eterminnt sine tht is the sie of the sstems we will e solving. Evluting eterminnt eterminnt of mtri: The eterminnt of the mtri A is given et A A eterminnt of mtri: The eterminnt of the mtri A is given et A A Although there re numer of ws to lulte eterminnt, for the ske of simpliit, we re onl giving one generl formul. The reer m referene prelulus, or liner lger ook for further evelopment.

3 Evluting eterminnt oils own to rememering the formuls given ove. Let s see how this works in the net emple. Emple : Fin the eterminnt of eh mtri..... Solution:. To fin the eterminnt of mtri, we simpl nee to follow the formul given ove. ) et Notie tht n es w to rememer the formul for eterminnt is to multipl long the igonls n sutrt those prouts. Just mke sure ou put the first entr in the mtri s the first prout in the eterminnt.. Agin, to fin the eterminnt we just use the formul, tht is, fin the ifferene of the prout of the igonls. We get ). To fin the eterminnt of mtri, we lso just ppl the formul. It s just tht the formul is muh lrger. However, there is pttern to it. Notie the first row is the numer in front of eh minor eterminnt with the pttern of signs) n eh minor eterminnt is wht we get simpl rossing out the orresponing row n olumn of the first row we re using. So, for our given mtri we get ) et Now we simpl hve to evlute the smller eterminnts n finish our omputtions.

4 ). Agin, we will follow the formul ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )) ) ) ) The eterminnt is ver powerful tool in mtries n n to numerous things. However, we re onl intereste in using the eterminnt to solve sstems of equtions. To o this we use something lle Crmer s Rule. This rule is nme fter th entur Swiss mthemtiin Griel Crmer. Crmer s Rule for Solving Sstems Consier the sstem Let the three eterminnts, n e efine s Then, if, the sstem hs unique solution of

5 Although Crmer s rule seems omplite, it s merel mtter of omputing the oeffiient mtri eterminnt n then omputing tht sme eterminnt where eh olumn is reple the onstnts in the sstem. Then, generting the frtions to get the solution. Let s look t it with the net emple. Emple : Solve the sstem using Crmer s Rule... Solution:. The first this we nee to o is etermine ll of the eterminnts, n formuls given ove to o so s follows.. We use the This gives ) ) ) ) ) ) So then we just nee to evlute So the solution is,. n n to get our solution. These give. As in prt. ove, we nee to etermine the eterminnts in orer to solve the sstem. This gives Clulting n we get So the solution is -, ). n

6 Although solving sstem with Crmer s Rule is not too iffiult, it is it more time onsuming n lor intensive to o sstems s we see net. Crmer s Rule for Solving Sstems Consier the sstem Let the four eterminnts,, n e efine s Then, if, the sstem hs unique solution of As ove with the se, this seems omplite. However, it s just mtter of lulting severl eterminnts n plugging the vlues in. Emple : Solve the sstem using Crmer s Rule.. Solution:. The first thing we nee to o here is to evlute ll of the orresponing eterminnts. We strt with the oeffiient eterminnt. ) ) ) )

7 Now we ompute the other eterminnts ) ) ) ) ) ) ) ) ) ) ) ) ) Now tht we hve tht, we simpl nee to etermine the solutions the formuls We get So the solution to the sstem is -,, ).. As we i in prt. ove, we nee to strt etermining ll of the eterminnts require for Crmer s Rule. We get

8 We will leve it to the reer to work through the etils of these eterminnts. It n e shown tht these eterminnts give us Plugging in gives us So the solution to the sstem is, -, -). The onl rel hng-up with Crmer s Rule is the unfortunte sitution in whih the enomintor eterminnt ens up s ero. If this is the se, Crmer s Rule oes not work. You woul hve to go k n solve the sstem using one of the other methos tht ou hve lerne for solving sstem.. Eerises Write the oeffiient mtri n the ugmente mtri of the following sstems Fin the eterminnt of eh mtri

9 ... Solve the sstem using Crmer s Rule

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