5 - Determinants. r r. r r. r r. r s r = + det det det

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Maude Francis
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1 5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow e sque.. Det I fo ll. 2. If mtx B s ote fom mtx A y tehgg two ows, the et B - et A. 3. The etemt s le wth espet to opetos o sgle ow. Tht s, suppose tht s,, 2,..., e vetos R (the ows of mtes), tht s y sl, tht s some tege fom to (the ume of ow). We eque tht et et et s s

2 et 2 2 et + + These thee popetes uquely efe the etemt (though o ot pove ttol fomul.) Note tht popetes 2 3 pemt us to tk the effet tht two of the elemety ow opetos hve o the etemt of mtx. How out the th ow opeto, g sl multple of oe ow to othe? It oes ot hge the etemt. Hee s how ths e see. Suppose tht mtx A hs two ows whh e equl. The tehgg those two ows yels the sme mtx. Fom popety 2 ove, et A - et A, hee, et A 0. Moe geelly, f A hs some ow whh s multple of othe of ts ows, et A 0. Fo suppose tht A ow s tmes ow j ( j ). Let B e the mtx whh s the sme s A exept ple of ow of A, B hs ow j of A (.e. ows j of B e oth j ). The, fom the seo pt of popety 3, et A et B. But we hve just see tht et B 0, so et A 0. Flly the usg the fst pt of popety 3, we see tht g sl multple of some ow to othe hges the etemt y to of the etemt of mtx of the type of B ove ( mtx wth some ow eg multple of othe.) But tht etemt s zeo, so the opeto oes ot hge the etemt.

3 We llustte usg these popetes to lulte etemt elow. Exmple Let A wte fo et A. We eue A keepg tk of the effet o the etemt. Mtx Detemt

4 Thus we hve et I 3, whh yels et A I oseg solutos fo systems of le equtos we fou t oveet to wte the systems s mtx equtos suh s Ax whee x e vetos, x olletg the ukows the ostts whh ompse the ght h se of eh equto. Ths ppoh poves wy to ollet lge mouts of t gves ovevew of the polem. Suppose tht the oeffet mtx A s sque mtx, sy A hs sze y so tht the system ossts of equtos ukows. We hve see tht the system hs uque soluto extly whe A ow eues to the etty mtx I. Coseg the 2 2 se, let A. Ue wht otos oes A eue to I 2 so tht

5 the system hs uque soluto? Assumg 0, the euto yels the followg mtes: 0 Fo ths to eue to I 2 the we must hve 0, so - 0. If 0, the must eessly e ozeo the euto looks lke 0 Hee we must hve 0 so - 0, o, s efoe, - 0. The ume - etemes the fom of the esult. How out the 3 3 se? Set S. e f g h Wth ppopte quttes eg ozeo the euto poees s te elow. e f g h 0 0 e f e h g eg * * f e X

6 Note tht the 2 2 mtx the uppe lefth oe s smply the 2 2 mtx we eue ove (wth ts 2,2 ety ewtte s sgle fto), tht stesks hve ee ple etes tht e of o osequee, tht euto to I 3 s epeet o X eg ozeo whee X eg f e h g ( ) eg f e h g 2 ( eg)( ) + ( e f)( h g) eg + eg + eh eg fh + fg ( ) ( ) ( + he hf + gf ge) ( ) ( ) + he ( f) + gf ( e) Hee, we get uque soluto extly whe the ume ( - ) + h(e - f) + g(f - e) s ot 0. All the othe ses ths 3 3 polem, e.g. whe 0 o - 0, yel ths sme esult. Pttes tht eg to emege fom these egg ses le to two ffeet fomuls fo omputg ths ume ssote wth the sque mtx A, ts etemt. Fst ote tht eh tem eh sum, oth the sum the 2 2 polem the sum fom the 3 3 se, s the pout of etes fom A, extly oe ety fom eh ow oe fom eh olum eh pout. Also the sum ossts of extly evey possle pout wth extly hlf of them lso hvg mus sg. Vewg the etemt ths wy les to the pemutto fomul tht we expl elow. Fom slghtly ffeet gle, the fomuls ve t fo the , mely - ( - ) + h(e - f) + g(f - e), espetvely, oth osst of sums of pouts wth the fst ftos eg the etes fom ptul ow. I the 2 2 se the mke up the fst ow of A: fo the 3 3 se t s the, h, g whh s the th ow of A. Exmg the tem

7 multplyg, t s ely eogze s the etemt of the 2 2 uppe lefth oe of A. I ft, eh of the quttes the petheses s (plus o mus) the etemt of the sumtx of A ote y eletg the ow olum of the ety whh multples tht goup. Ths s lso tue of the 2 2 se f we tepet the etemt of mtx s eg ts loe ety. The eusve fomul follows fom ths vewpot. Reusve Fomul fo the Detemt If A s mtx, the set et A, the oly elemet of A. Suppose tht A s mtx wth >. Let A ( j ), fo the,j th ety of A, efe the mo of j, eote M j, to e the sque mtx fome y eletg the th ow the j th olum of A. The ofto of j, C j, s the etemt of M j. The et A C - 2 C C (-) + C. Ths s lle the ofto expso of A out ts fst ow. Exmple Cose the mtx A. The ofto expso of A out the fst ow yels the etemt of A s et - (-3) et + (-) et -7et Demosttg the euso the fomul we ompute (g usg expso out the fst ow) the ofto ssote wth the, ety 5.

8 et et - (-6) et + 20 et (-2) + 6(0) + 20(-2) Fllg the vlues fo ll these oftos we get et A 5(-36) + 3(-20) - (0) - 7(-20) Note tht the ofto of the,3 ety (-) s zeo se we oul exp out the seo ow usg the ft stte elow. I ft, ou tsk woul hve ee smple h we expe the ogl mtx A out ow 3. Wth the emg etes ll g zeos, we woul hve just gotte (-) et 2(-450) se ths s the mtx whose etemt we ompute the pevous exmple. Ft: The etemt e ompute y expg out y ow o olum,.e. + + j j ( ) C j j ( ) Det A the fst sum s the expso out ow, j C j j the seo the expso out olum j. Pemutto Fomul fo the Detemt Ag suppose tht A ( j ) s mtx. Let σ e pemutto of the set {, 2,...,},.e. {, 2,...,} {σ(), σ(2),...σ()}. (Wht we hve s σ s s jeto, oe-to-oe, oto futo fom {, 2,...,} to tself.) Fo y pemutto σ, we efe the sg of σ, sg σ, y outg the ume of moves eque to ege the sequee σ()σ(2)...σ() oe to fom the sequee If ths ume s eve the sg σ, whle f ths ume s o set sg σ -.

9 Flly ltete, equvlet efto of etemt s et A sg( σ) σ () 2σ ( 2 )... σ ( ) whee the sum s tke ove ll the pemuttos of {, 2,...,}. Se thee e! ( ftol) pemutto o egemets of {, 2,...,}, Stg efes to ths s the Bg Fomul. Exmple Let A The pemuttos of {, 2, 3} e 23, 23, 32,32, 23, 32. The fst thee of these e eve, the lst thee o. Thus, usg the pemutto fomul, et A @6@0 + 3@7@8 + 4@5@9-2@7@9-3@5@0-4@6@8 30 Appltos The ft tht et A 0 ws eessy suffet oto fo systems wth oeffet mtx A to hve uque soluto hols fo y whh we eo the followg Theoem. Let A e mtx.. The equto Ax hs uque soluto (fo y R ) f oly f et A A s vetle f oly f et A 0. Beses sglg the exstee of vese fo A, the etemt s use fomul fo A -. A - C T whee C (C j ) s the mtx of oftos of A ( the, j th ety of C, et A C j, s the,j th ofto of A.) So A - ossts of the tspose of the mtx of oftos of A

10 tmes the sl. et A Colletg some popetes of the etemt we ote tht fo y mtes A B y sl :. et (AB) et A et B, 2. et A et A, 3. et A T et A, 4. et A - et A. Iopotg eftos esults fom pevous setos, we gthe ume of otos tht e equvlet fo sque mtes the theoem elow. Theoem Suppose tht A s mtx. The followg e equvlet.. The etemt of A s ot A s vetle. 3. A eues to the etty mtx, I. 4. The equto Ax hs uque soluto fo eh R. 5. The equto Ax 0 hs oly the tvl soluto. 6. The k of A s. 7. The meso of the ow spe of A s. 8. The meso of the olum spe of A s. 9. The meso of the ull spe of A s The olums of A e lely epeet.. The olums of A sp R. 2. The ows of A e lely epeet. 3. The ows of S sp R.

SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS

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