Walk Like a Mathematician Learning Task:

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1 Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics inclu solvin systms of qutions, finin th r of trinls ivn th coorints of th vrtics, finin qutions for rphs ivn sts of orr pirs, n trminin informtion contin in vrtx rphs. In orr to rss ths typs of problms, it is ncssry to unrstn mor bout mtrix oprtions n proprtis; n, to us tchnoloy to prform som of th computtions. Mtrix oprtions hv mny of th sm proprtis s rl numbrs. Thr r mor rstrictions on mtrics thn on rl numbrs, howvr, bcus of th ruls ovrnin mtrix ition, subtrction, n multipliction. Som of th rl numbr proprtis which r mor usful whn consirin mtrix proprtis r list blow. Lt, b, n c b rl numbrs ADDITION PROPERTIES MULTIPLICATION PROPERTIES COMMUTATIVE + b = b + b = b ASSOCIATIVE ( + b) + c = + (b + c) (b)c = (bc) IDENTITY Thr xists uniqu rl Thr xists uniqu rl numbr zro, 0, such tht numbr on,, such tht + 0 = 0 + = * = * = INVERSE For ch rl numbr, For ch nonzro rl numbr thr is uniqu rl, thr is uniqu rl numbr - such tht numbr such tht + (-) = (-) + = 0 ( ) = ( ) = Mthmtics : Mtrics Richr Woos, Stt School Suprintnnt July 205 P 2 of 70

2 Gori Dprtmnt of Euction Th followin is st of mtrics without row n column lbls. Us ths mtrics to complt th problms. D = 2 0 E = F = 5 2 G = 0 2 H = = 4 2 I 0 0 J = K = 0 2 L = In this problm, you will trmin whthr mtrix ition n mtrix multipliction r commuttiv by prformin oprtions n comprin th rsults. D + J = J + D?. Dos [ ] [ ] [ ] [ ] b. Dos [ E ] [ K ] = [ K ] + [ E] +? c. Is mtrix ition commuttiv? Why or why not?. Dos [ D ] [ J ] [ J ]*[ D] * =?. Dos [ E ] [ H ] [ H ]*[ E] * =? f. Is mtrix multipliction commuttiv? Why or why not? Mthmtics : Mtrics Richr Woos, Stt School Suprintnnt July 205 P of 70

3 Gori Dprtmnt of Euction 2. Ar mtrix ition n mtrix multipliction ssocitiv? D + J + L = D + J + L?. Dos ([ ] [ ]) [ ] [ ] ([ ] [ ]) b. Is mtrix ition ssocitiv? Why or why not? c. Dos ([ D ] [ J ]) *[ L] [ D] *([ J ]*[ L] ) * =?. Is mtrix multipliction ssocitiv? Why or why not?. Is thr zro or intity, 0, for ition in mtrics? If so, wht os zro mtrix look lik? Provi n xmpl illustrtin th itiv intity proprty of mtrics. 4. Do mtrics hv on or n intity, I, for multipliction? If so wht os n intity mtrix look lik; is it uniqu; n, os it stisfy th proprty * = * =?. Multiply [ E ]*[ I ] n [ I ]*[ E]. Dscrib wht you s. 5. Fin [D]*[G] n [G]*[D]. Dscrib wht you s. D n G r cll invrs mtrics. In orr for mtrix to hv n invrs, it must stisfy two conitions.. Th mtrix must b squr mtrix. 2. No row of th mtrix cn b multipl of ny othr row. Both D n G r 2x2 mtrics; n, th rows in D r not multipls of ch othr. Th sm is tru of G. Th nottion normlly us for mtrix n its invrs is D n D - or G n G -. Mthmtics : Mtrics Richr Woos, Stt School Suprintnnt July 205 P 4 of 70

4 Gori Dprtmnt of Euction 6. Multiply G by n look t th rsult. Cn you s ny rltionship btwn D n th rsult? Th followin formul cn b us to fin th invrs of 2x2 mtrix. Givn mtrix A whr th rows of A r not multipls of ch othr: b A = thn A - = c bc c b. Fin th invrs of mtrix J from th mtrics list bov. Vrify tht J n J - r invrss. b. Now your tchr will show you how to us tchnoloy to fin th invrs of this mtrix. 7. A uniqu numbr ssocit with vry squr mtrix is cll th trminnt. Only squr mtrics hv trminnts. A. To fin trminnts of 2x2 mtrics by hn us th followin procur. b trminnt(a) = t(a) = A = = bc c. Fin D b. Fin J c. Cn you fin F? Why or why not? B. On wy to fin trminnts of x mtrics us th followin procur: ivn mtrix b c b c b B = f, rwrit th mtrix n rpt columns n 2 to t f. h i h i h Now multiply n combin proucts ccorin to th followin pttrns. Mthmtics : Mtrics Richr Woos, Stt School Suprintnnt July 205 P 5 of 70

5 Gori Dprtmnt of Euction b h c f i b h n b h c f i b h = t(b) = i + bf + ch c fh bi.. Fin E b. Fin K c. Now your tchr will show you how to us tchnoloy to fin th trminnts of ths mtrics. 8. Th trminnt of mtrix cn b us to fin th r of trinl. If (x, y ), (x 2, y 2 ), n (x, y ) r vrtics of trinl, th r of th trinl is x y Ar = x2 y2. 2 x y. Givn trinl with vrtics (-, 0), (, ), n (5, 0), fin th r usin th trminnt formul. Vrify tht r you foun is corrct usin omtric formuls. Mthmtics : Mtrics Richr Woos, Stt School Suprintnnt July 205 P 6 of 70

6 Gori Dprtmnt of Euction b. Suppos you r finin th r of trinl with vrtics (-, -), (4, 7), n (9, -6). You fin hlf th trminnt to b n your prtnr works th sm problm n ts Aftr chckin both solutions, you ch hv on your work corrctly. How cn you xplin this iscrpncy? c. Suppos nothr trinl with vrtics (, ), (4, 2), n (7, ) ivs n r of 0. Wht o you know bout th trinl n th points?. A rnr is tryin to fin trinulr r bhin his hous tht ncloss 750 squr ft. H hs plc th first two fnc posts t (0, 50) n t (40, 0). Th finl fnc post is on th proprty lin t y = 00. Fin th point whr th rnr cn plc th finl fnc post. Mthmtics : Mtrics Richr Woos, Stt School Suprintnnt July 205 P 7 of 70