The total number of permutations of S is n!. We denote the set of all permutations of S by

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1 DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote the set of ll permuttios of S y Exmple. S {,, } S {,,,,, }! 6 permuttios Def: permuttio jjj j of S is sid to hve iversio if lrger iteger, sy jq, precedes smller oe, sy, jr. permuttio is clled eve permuttio if the totl umer of iversios i it is eve, or odd if the totl umer of iversios i it is odd. Exmple. If, there re! eve d! odd permuttios i S. ) S hs permuttio:, which is eve ( o iversio ) ) S hs permuttios:, which is eve (o iversio) d, which is odd ( iversio ) ) I the permuttio i S, the totl umer of iversios is 5. Thus, is odd permuttio. ) I the permuttio 5 i S5, the totl umer of iversios is 5, d 5 hs 6 iversios. ij e x mtrix. The determit of, deoted y det( ) or, is defied y ( ± ) j j j S j where the summtio is over ll permuttios jjj j of the set S {,,, }. The sig is tke s ( + ) whe the permuttio is eve d ( ) whe it is odd. Def : Let { } Illustrtio: ) ; S {, } : thus 7

2 ) B ; S {,, }: thus B Exmple. Let 9 6. The Bsket Rule for x d x mtrices: B Remrks:. Determits re defied oly for squre mtrices. The determit of o squre mtrix is udefied d therefore does ot exist.. det(sclr) sclr. Def: Let { ij } e x mtrix. Let M ij e the ( ) x ( ) sumtrix of otied y deletig the ith row d jth colum of. The determit M ij is clled the mior of ij. Def: Let { ij } e x mtrix. The cofctor, ij, of ij is defied s ij ( ) i+j M ij. (o. of cofctors ) 8

3 Determit, Def. : (Usig cofctor expsio) Let { } ij e x mtrix. The ij ij j i ij j ij, for y i ( ) i+j M ij (expsio of out the ith row) ij, for y j or (expsio of out the jth colum) ij ( ) i+j M ij i Note: ) The cofctor expsio is used recurretly whe is lrge, i.e., ech M ij is expded y the sme procedure. ) Expsio out y row will produce determit which is the sme s whe expsio is doe out y colum. ) Expsio should e doe out the row/colum which hs the lrgest o. of zeros. Computtios:. First Order Determit, R Exmple. 7, 8, 5 B. Secod Order d Third Order Determits use Bsket Rule Exmple.5 ) 6 5 ) 9 B, the, the B 9

4 5 ) D , the D ) F 5 6, the F C. Higher Order Determits - use cofctor expsio Exmple.6 ), ) B 6, B. Properties of Determits The followig results re theorems (for proofs, see Serle). Let { } ij e x mtrix ) '. sice expsio out the row is equivlet to expsio out the colum

5 ) If rows (cols) of re the sme, the. sice if hs rows which re the sme, we c expd y miors so tht x miors i the lst step of the expsio re from equl rows. The for ll miors ) If oe row (col) of mtrix is multiple of other row (col), the determit is. fctor out the costt (multiplier) to produce determit with rows (cols) the sme ) If hs zero col (row), the. expd out tht col (row) ii. i use cofctor expsio recurretly log the row/col with the most s 5) If is trigulr mtrix, the 6) If is digol mtrix, the ii. i 7) Whe ozero sclr λ is fctor of row (col) of, the it is lso fctor of, i.e., λ with λ fctored out of row (col) Exmple ) If λ is sclr, λ λ. 9) If is skew symmetric, d is odd, the. ' ( ), is odd - iff. ) If d B re squre mtrices d re of the sme order, the B B. 5

6 ) For d B squre mtrices of the sme order, B B. sice B B. ) k k, where k is positive iteger. ) If is orthogol, the ±. sice I d I ' ± ) If is idempotet, the,. sice, 5) For d B squre mtrices of the sme order, if B I, the d B. sice B B I d B O 6) For d B squre mtrices of the sme order, I B. 7) If d B re squre mtrices, ot ecessrily of the sme order, the O O B B. 8) If, B d C re mtrices of order x, the C O B B.. Elemetry Row Opertios Def: elemetry row (col) opertio o mtrix 5 mx is y oe of the followig opertios: ) Type I opertio: iterchge row (col) i d row (col) h. ) Type II opertio: multiply row (col) i y c. c) Type III opertio: dd multiple of row (col) i to row (col) h, i h.

7 ... How Type I elemetry row or colum opertio c e doe usig mtrix opertios: Let 6. Wht we will do is iterchge the the d d rd row of the mtrix. To do 7 tht, we will post multiply to the mtrix E. If we do the multiplictio, we will get B E 7 6. Let us switch the d colum d the rd colum of. To do this, we will pre multiply E. Wht we will get is B E 6. 7 Exercise (ssigmet):. Usig the mtrix ove, switch the st row with the d row, y defiig ew mtrix to pre multiply to.. Usig the mtrix ove, switch the st colum with the rd colum, y defiig ew mtrix to post multiply to... B. How Type II elemetry row d colum opertios re doe: Pre multiply the mtrix G to. Post multiply the mtrix G to. Wht re c the results? Wht hve you oticed?.. C. How Type III elemetry row d colum opertios re doe: Exmples:. ddig the first colum of to its third colum: 6 7 5

8 . ddig the first row to the third row: 6 7 Exercise (ssigmet):. dd the secod row of to its third row d show the mtrix multiplictio tht does this.. dd the secod colum of to its first colum d show the mtrix multiplictio tht does this. dditiol Results o Determits: Let e x mtrix ) If mtrix B is otied from y iterchgig rows (cols) of, the B. Exmple: Solve for E, d E. Show tht E E E?. Wht is the for of ) If B is otied from y multiplyig row (col) of y rel o. k, the B k. Exmple: Solve for G, G d G. Show tht G G d G G. Wht is the form of G? ) If B is otied from y ddig multiple of row (col) i to row (col) h, i h, the B. Exmple.8 ) Let d B. The d B 5

9 ) If ut ) Let 6. Digol Expsio 6, the, the Def: Deletig y r rows d r cols from squre mtrix of order leves sumtrix of order ( r). The determit of this sumtrix is mior of order ( r), or ( r) order mior. Def: pricipl mior is mior whose digol elemets re coicidet with the digol elemets of the origil mtrix. mtrix, sy X, c lwys e expressed s the sum of two mtrices, oe of which is digol mtrix, i.e., X + d + D where { } ij for i, j,,, d D is digol mtrix of order. The determit of X c the e otied s polyomil of the elemets of D. Cosider the mtrices { ij }, i, j, d D dig{ d, d }. The + D I similr fshio, it c e show tht + d + d Cosidered s polyomil i the d s, we c see tht i) is the coefficiet of the product of ll the d s. ii) iii) digol elemets of re the coefficiets of the d degree terms i the d s. d order pricipl miors of re coefficiets of the st degree terms i the d s. iv) is the term idepedet of d s. 55

10 This method of expsio is kow s expsio y digol elemets or simply digol expsio. This method of expsio is useful o my occsios ecuse the determitl form pricipl miors re zeros, the expressio + D occurs quite ofte, d whe is such tht my of its + D y this method is gretly simplified. Exmple.9 ) Let X the we hve X + D ) X 6 6 Remrk: If D is sclr mtrix, i.e., the d i s re equl, the + D The geerl digol expsio of determit of order, + D cosists of the sum of ll possile products of the d i s tke r t time for r,,,,,, ech product eig multiplied y its complemetry pricipl mior of order ( r) i. By complemetry pricipl mior i is met the pricipl mior hvig digol elemets other th those ssocited i + D with the d s of the prticulr product cocered. Whe ll the d s re equl, the expressio ecomes where () d () i + D d tr i ( ) tr i is the sum of the pricipl miors of order i of. By defiitio, tr o () tr. i 56

11 .5 Sums d Differeces of Determits ) I geerl, + B + B Exmple. + B ) I geerl, B B Note: If is x mtrix d B is m x m mtrix, ± B is defied ut ± B is ot. ) If { ij } d B { ij } re x mtrices tht re ideticl for ll elemets except for correspodig elemets i the kth row, d if C { cij } is x mtrix, the + B C, where cij ij, except i the kth row, i which ckj kj + kj, j,,,. Proof: Exercise REDING SSIGNMENT: Red Chpter 5: Iverse Mtrices, pp Do the Exercises o pp 8 5 (for iverse mtrices) d pp. to 8 (for determits) 57

12 CHPTER PPENDIX : EVLUTING DETERMINNTS USING EXCEL ND SS We c lso use Excel d SS to simply get the determits, without goig to the troule of usig the sket rule, the cofctor expsio methods, or digoliztio. I Excel, we simply use the MDETERM(rry) fuctio. Exmple, we hve 9 6 To solve for, we type the mtrix d use the fuctio o other cell. By the output,.. I SS, this is simply the DET() fuctio: SS Code: {, 9, 6 }; B Det(); prit B; ru; SS Output: The SS System B

13 CHPTER PPENDIX B: HELPFUL MTRIX FUNCTIONS IN SS Recetly goig through the IML Lguge Referece i SS, I d like to give you some of the fuctios tht my help you i usig SS for mtrix lger. Geerlly, these fuctios will show you how to mke the specil mtrices tht we use i clss, such s the idetity mtrix, mtrix of oes, d digol mtrices. I ll throw i more fuctios i other ppedices i lter chpters.. BS(mtrix) it gives the solute vlues of the elemets of the origil mtrix. Exmple: { -, -, -, - }; s_ s(); prit s_; ru; BS_ BLOCK(mtrix <,mtrix,,mtrix5>) it crete mtrix with sumtrices rrged digolly. Exmple: {, } ; {6 6, 8 8} ; clock(,); prit c; ru; C DIG(rgumet) if the rgumet is mtrix, it returs with the digol elemets. If the rgumet is vector, the it gives digol mtrix with the elemets o the rgumet. Exmple: {, }; cdig(); { }; ddig(); prit c d; ru; 59

14 C D. EXP(mtrix) clcultes the expoetil t ech elemet of the mtrix Exmple: { }; exp(); prit ; ru; I(dimesio) it gives the idetity mtrix of the give dimesio Exmple: I(); prit ; ru; 6. J(row <, col <, vlue > > ) it gives mtrix with commo vlue Exmple: j(); rj(5,,'xyz'); kj()*; prit r k; ru; B R K xyz xyz xyz xyz xyz xyz xyz xyz xyz xyz 6

15 7. T(mtrix) this is other fuctio tht gives the trspose of the origil mtrix rgumet. Exmple: x{, }; yt(x); prit x y; ru; X Y 8. XMULT(mtrix, mtrix) lso performs mtrix multiplictio ut with greter ccurcy. Exmple: x{, }; yt(x); zxmult(x,y); prit x y z; ru; X Y Z 5 5 6