# (2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

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1 . DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion, somtim w wnt th volumn o tht. Thn w will in tht trminnt is th right vlu to work out. Historilly spking, trminnt trmins whn os systm o qution with n unknowns n n qulitis hv uniqu solution. In our mtrix lngug, trminnt trmins whn mtrix is invrtil. You will in out tht trminnt givs you st wy to trmin whn mtrix is invrtil n wht is th invrs. In n 3 3 it is st, ut in 4 4 or mor imnsion ss, th wy w lrn or might th most st wy. W not th st o ll m n mtris ovr F y M m n Thorm. Thr xists untion in on M n n. Tht is ssign vry n n mtrix ovr F vlu in F. whih sitisying: () I w swith ith n jth rows o B, thn th vlu oms to opposit. nmly (S ij B) (B) () I w multipli row o B y λ, thn th vlu is lso multipli y λ(hr lm oul 0). nmly (M i (λ)b) λ(b) (3) I w ny slr tims jth row to ith row o B, thn th vlu osn t hng. nmly (A ij (λ)b) (B) Thn this kin o untion xists, n ny two suh untion is ir y slr multipl. As ny suh two untions ir y slr multipl, this mns whnvr w ins th vlu o unit mtrix, thn th untion is uniquly trmin. This lt us l to mk ollowing inition. Dinition.. Th trminnt is uniqu untion on n n mtris, whih ssign vry n n mtrix M ovr F vlu in F. not y M or t(m) whih sitisying: () I w swith ith n jth rows o mtrix, thn th trminnt oms opposit. xpliitly: i i i3 in j j j3 jn n n n3 nn j j j3 jn i i i3 in n n n3 nn

2 () I w multipli ny row y λ, thn th trminnt will lso multipli y λ(hr lm oul 0). xpliitly: n n λ i λ i λ in λ i i in n n nn n n nn (3) I w ny slr tims jth row to ith row o B, thn th vlu osn t hng. xpliitly: i + λ j i + λ j i3 + λ j3 in + λ jn j j j3 jn n n n3 nn i i i3 in j j j3 jn n n n3 nn (4) Th trminnt or unit mtrix is In othr wors, trminnt is th uniqu untion tht sitisying t(s ij B) t(b); t(m i (λ)b) λ t(b); t(a ij (λ)b) t(b); t(i n ) This inition is silly wht is trminnt, n ll othr proprtis o trminnt n ll u rom its uniqunss Comput th trminnt o ollowing mtris W pro s ollowing: ; r + ( ) r r + 3 r ( ) ( )

3 Proposition. I on row is linr omintion o othr rows, thn th trminnt is 0 This is skill whn you n s tht, ut on t worry i you n t. you n lult ll trminnt just y row trnsormtion.(lso olumn trnsormtion is vli, w will tlk out it in th nxt w propositions) Clult 4 6 Som on n s tht son lin is twi o th irst lin, so th trminnt shoul 0. But on n lso lult: 4 6 r + ( ) r 0 * n nything 0 nywy, i on row o trminnt is 0, th trminnt is just 0 4 Clult Som on n s tht 3r row is th irst lin plus iv tims th son row, so th trminnt shoul 0. But on n lso lult: r 3 + ( ) r r 3 + ( 5) r 0 us th lst row o trminnt is 0 row Proo. Th irst proprty is lrly rom th xiom. 3

4 Proposition. Th trminnt or igonl mtrix is th prout o ll igonl lmnts Proo nn nn nn 33 nn... Proposition.3 Th trminnt o th uppr tirngulr mtrix is th prout o igonl lmnts, tht is 3 n i3 in nn 33 nn Proo. W n trnsr uppr tringulr mtrix to igonl mtrix y only row ition, with thir igonl lmnts kpt. But row ition os not hng th trminnt, so it is th sm with th trminnt o igonl mtrix 4

5 4 5 Clult 6 9 Answr: 9 8 Proposition.4 Th mtrix M is invrtil i n only i t(m) 0 Proo. I t(m) 0, thn or ny lmntry row trnsormtion o M, th trminnt is 0. in othr wors, or ny invrtil mtrix P, t(p M) 0, w hoos P suh tht PMr(M). so y this prtiulr hoi, w know t(r(m)) 0, us r(m) is uppr tringulr, so trminnt shoul th prout o igonl. tht mns thr r 0 in igonl. so M is not o ull rnk. I t(m) 0 thn or ny lmntry row trnsormtion o M, th trminnt os not qul to 0 us h tim only ir y non-zro slr multipl. Tht is, or ny invrtil mtrix P, t(p M) 0. so hoos P suh tht PMr(M), so t(r(m)) 0, so ll lmnt o r(m) r non zro, tht mns r(m) is n unit mtrix. so M is o ull rnk, so M is invrtil. Proposition.5 t(ba) t(b) Proo. I on o A or B is not invrtil, thn oth si o th qution is 0, so tru. I A is invrtil, thn w onsir th untion (B) t(ba) Lt s hk (S ij B) t((s ijb)a) t(s ij(ba)) t(ba)) (B). An w n lso us th sm mtho to hk (M i (λ)b) λ(b) n (A ij (λ)b) (B) n (I n ). So (B)t(B). Thn t(ba) t(b), tht is t(ba) t(b) Proposition.6 Th trminnt lso sitisis olumn trnsormtion proprty, tht is t(bs ij ) t(b); t(bm i (λ)) λ t(b); t(ba ij (λ)) t(b); Proo. By prvious proposition, trminnt is multiplitiv, so t(bs ij ) t(b) t(s ij ) t(b); 5

6 t(bm i (λ)) t(b) t(m i (λ)) λ t(b); t(ba ij (λ)) t(b) t(a ij (λ)) t(b); This mns whn w lult trminnt w n oth us olomn n row trnsormtion. Proposition.7 t(a T ) Proo. Consir th untion (B) t(b T ), w hv (S ij B) t((s ij B) T ) t(b T Sij T ) t(b T ) t(sij T ) t(bt ) (B) n w n hk (M i (λ)b) λ(b); (A ij (λ)b) (B); (I n ) y th sm pross, tht is (B) t(b). this mns t(b) t(b T ) Corollry. Th trminnt o lowr tringulr mtrix is th prout o ll ntris in igonl. Without multiplying two mtrix, omput t( Answr: t( ) t( ) t( 3 8 ) ) Nxt, w giv som lultion mtho or son orr trminnt n thir orr trminnt. Bor w pro, w illustrt tht thr is on mor importnt proprty o trminnt tht worth using: Proposition.8 n i + i i + i i + in n n nn n i i i n n nn + n i i i n n nn Proo. This proprty oul lso prov y uniqunss o trminnt. For A ( ij ) i n onsir th untion 6 j n

7 (A) n i + i i + i i + in n n nn n i i i n n nn As w hk th thr proprtis o, w shoul pro slightly tk r o. W omit proo hr, Th rr oul o tht, goo xris. Proposition.9 Proo. + Proposition.0 i + g + h g i h Proo. W us th sm mtho s w i or, until w n gt n uppr tringulr, lowr tringulr or lok uppr or lowr tringulr mtrix g i h + h i g h g i i h g + i g h i h + g i g + h i + g + h g i h 7

8 Clult th trminnt o 5 7 Answr: Clult th trminnt o Answr: Clult th trminnt o 3 9 Answr: I wnt you to prti proprty o row ing

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