Ejercicios Resueltos de Determinantes Con Soluciones Infinitas, Unicas, y Vacias

 

 

WHIFKF ; JKV_ING^

 

KABGO_K AIHGKA Ihb. Ugrþhink \rckûc _êcs J^n. \`F(n) FG\K_VKJGHVC FG NIGHNIK^ GPKNVK^ - G^\G 

 

WHIFKF Hrc. ;

   

Kntivifkf fg kprghfizkdg #6 Gdgrninics ncrrgspchfightgs ka tgjk 64 ^istgjk fg Gnuknichgs Aihgkags Hcjorg fga gstufikhtg  Nkrrgrk 

Jktgc Vkpik Ihbghigrêk Nivia

H_N 

>077

Hcjorg fga prcegscr 

Ihb. Ugrþhink \rckûc _êcs J^n. \`F(n)

Ihfinknichgs4 - 

\krk rgkaizkr acs gdgrninics fg Jktrings agk acs krn`ivcs fg nchtghifcs fg aks naksgs  Hrc. ; y ?, ksê ncjc acs vifgcs fg fg kpcyc.

- 

_gkaizkr ? gdgrninics fg nkfk suotgjk. Gstcs sg ihfinkh gh ga ^êakous fg ak ksibhkturk.

Gd4 Jktrings nchtighg 6 suotgjks, Fgtgrjihkhtgs 6 suotgjks, gtn. -  \krk fgskrrcaakr acs gdgrninics sg rgncjighfk `kngr usc fg uh gfitcr fg eþrjuaks. - 

Acs gdgrninics pcfrëh nchtghgr ncjoihkniþh fg vkrics suotgjks y/c prcpigfkfgs.

- 

^g rgncjighfk jghnichkr aks prcpigfkfgs kpainkfks gh acs gdgrninics.

- 

Fgog ghtrgbkr uh krn`ivc pcr nkfk tgjk t gjk gh ecrjktc Rcrf c \FE, nuyc hcjorg fgog tghgr ga sibuightg ecrjktc4 H_NS HVSKpgaaifc;SKpgaaifc?

Fchfg4  H_N 4 Gs ga H_N fchfg sg ghnughtrk jktrinuakfc/k  HV 4 Gs ga hôjgrc fg tgjk ihfinkfc gh ga sêakous. Gh ga nksc fg Jktrings gs ga ;  Kpgaaifc;4 Kpgaaifc pktgrhc  Kpgaaifc?4 Kpgaaifc jktgrhc Gd.4 >>>>S;S\rckûcS_ics.pfe >>>>S;S\rckûcS_ics.pfe - 

^uoir ga krn`ivc gh ak nkrpgtk ncjpkrtifk gh Bccbag Frivg gh ga pkrnika ncrrgspchfightg.

- 

Egn`k aêjitg fg ghtrgbk4 Fcjihbc fg ak sgjkhk qug sg fic ga ôatijc suotgjk.

6.;  Fgeihinichgs y prcpigfkfg prcpigfkfgs. s. Gdgrninic ;4 - 

Fgjcstrkr ga ghuhnikfc Nukhfc uh sistgjk `cjcbéhgc tighg scauniþh ôhink, acs ‑

vkacrgs scauniþh fg aks ihnþbhitks sch ibuka k xx;;++ 6x?x?++>xx>> ++ ?>xx66 :: 6; 5>?  ∅x6

. ^ih gjokrbc, ka tghgr uhk

ncaujhk huak, pcr prcpigfkfgs sg skog qug su fgtgrjihkhtg gs ibuka k : ;

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Gh oksg k uh sistgjk hc `cjcbéhgc nch scauniþh ôhink. _gscavgr ga sistgjk pcr ga   jétcfc fg Nrkjgr y ncjprcokr.

 =

 

6

 

 

WHIFKF Hrc. ;

 

;)  Fkfk ga sistgjk sg prcngfg k fgeihir ak jktriz K, y ga nchduhtc O

7

 

WHIFKF Hrc. ;

   

?)  ^g prcngfg k sknkr ga fgtgrjihkhtg fga sistgjk, s istgjk, pkrk pcfgr rgscavgr pcr ga jétcfc fg Nrkjgr, utiaizkhfc ga jétcfc fg Ncekntcrgs >)  Fg k`ê, sg prcngfg sknkr ga fgtgrjihkhtg

∅x pcr ga jgtcfc fg ncekntccrrgs.

;)  Fgspués, sg prcngfg sknkr ga fgtgrjihkhtg

∅y pcr ga jgtcfc fg ncekntcrgs. ∅z pcr ga jgtcfc fg ncekntcrgs.

?)  \cr nchsibuightg, sg prcngfg sknkr ga fgtgrjihkhtg

5

 

 

WHIFKF Hrc. ;

 

;)  Eihkajghtg, sg prcngfg sknkr ga fgtgrjihkhtg

∅e pcr ga jgtcfc fg ncekntcrgs.

?)  \krk ghnchtrkr ga vkacr scauniþh fg aks a ks ihnþbhitks fga sistgjk sg prcngfg k fivifir ga fgtgrjihkhtg fg ak vkrikoag pkrk ga fga sistgjk, pkrk ksê cotghgr ga g a vkacr fg ak ihnþbhitk.

2

 

WHIFKF Hrc. ;

   

  . ^g fgog ghnchtrkr   ;)  \krk ncjprcokr ga ghuhnikfc fkfc ?)  Ncjc yk cotuvijcs ga vkacr fga fgtgrjihkhtg fga sistgjk, prcngfgjcs k ghnchtrkr ak jktriz Kfduhtk fg K  ‑

x : ∟O

‑

 ∟

;)  Fgspués, sg prcngfg k trkhspchgr ak jktriz kfduhtk fg K cotghifk khtgricrjghtg. ?)  ^g prcngfg k fivifir pkrk su fgtgrjihkhtg, pkrk pcstgricrjghtg pcstgri crjghtg juatipainkrac pcr ga nchduhtc O

0

 

WHIFKF Hrc. ;

   

;)  Whk vgz, sg juatipaiqug pkskjcs k ncjpkrkr n cjpkrkr ak jktriz rgsuatkhtg fga sistgjk pcr , nch ak rgspugstk fga jétcfc fg Nrkjgr.

x : ∟O

‑

‚

?)  Ka vgrieinkr qug aks rgspugstks sch aks jisjks. Ga ghuhnikfc sg vgrieink qug si nujpag.

-;6 x : ;;9;57795 >;( 7 )

 

❕ 

^g nujpag ga ghuhnikfc, fgoifc k qug nukhfc uh sistgjk sis tgjk hc `cjcbéhgc c `cjcbéhgc tighg scauniþh ôhink, sg pugfg cotghgr acs vkacrgs scauniþh s cauniþh fga sistgjk, ka juatipainkr su ihvgrsk pcr ga nchduhtc O. 

6.?  Jétcfcs fg rgscauniþh fg uh sistgjk fg j gnuknichgs aihgkags nch h ihnþbhitks.

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Gdgrninic ; Fkfc ga sibuightg sistgjk. Khkaizkr si ga sistgjk tighg scauniþh ôhink, scaunichgs iheihitks, c si hc tighg scauniþh. Whk vgz fgtgrjihkfk ak rgspugstk, rgscavgr ga sistgjk.

>?xx -+y+5yy ++ 966zz :: ? -;5 96 y + 6z : ? ? -> 6 ?   =

 

9

 

 

WHIFKF Hrc. ;

 

;)  ^g prcngfg k ghnchtrkr acs vkacrgs ncrrgspchfightgs fg ak jktriz K y fg su jktriz kjpaikfk. ?)  ^g prcngfg k ghnchtrkr ga rkhbc fg aks fcs jktrings pkrk vgrieinkr qug ga sistgjk tghbk scauniþh ôhink.

;<

 

WHIFKF Hrc. ;

   

>)  Ka tghgr ga jisjc rkhbc ak jktriz jktri z ncjc ga fg su kjpaikfk, sg prcngfg k ghnchtrkr acs vkacrgs scauniþh fga sistgjk. 6)  Ka `kogr ghnchtrkfc ga rkhbc pcstgricrjghtg, hcs ekniaitk ghnchtrkr z  pcr ga jétcfc fg Bkuss. ‑

‚

7)  Whk vgz qug sg cotuvc ga vkacr fg z , sg rggjpakzk gh ak gnukniþh pkrk cotghgr ga vkacr fg y   ‑

‑

‚

‚

;)  Nch acs vkacrgs fg y  y fg z  sg prcngfg k rggjpakzkr gh ak gnukniþh pkrk cotghgr ga ‑

‚

‑

‚

>;0 >+5?? ? < -?-?;< x : ;;6- +>; > ?? +5-?<

vkacr fg x  y ksê cotghgr ga sistgjk scauniþh ‑

‚

 

;;

 

WHIFKF Hrc. ;

   

Gdgrninic ? - 

Fkfc ga sistgjk, ghnchtrkr acs sibuightgs aitgrkags4 aitgrkags 4



 

i-]-k>:?k cs5k 6k ?k  

k)  Fgtgrjihkr aks rgspugstks fga sistgjk

;)  ^g prcngfg k ghnchtrkr acs vkacrgs ncrrgspchfightgs fg ak jktriz K y fg su jktriz kjpaikfk. ?)  ^g prcngfg k ghnchtrkr ga rkhbc fg aks fcs jktrings pkrk vgrieinkr qug ga sistgjk tghbk scauniþh ôhink. >)  Ka tghgr ga jisjc rkhbc ak jktriz jktri z ncjc ga fg su kjpaikfk, sg prcngfg k ghnchtrkr acs vkacrgs scauniþh fga sistgjk.

;?

 

 

WHIFKF Hrc. ;

 

;)  Ka `kogr ghnchtrkfc ga rkhbc fg ak jktriz, nksi fgdkjcs ak rgspugstk fg ak vkrikoag (x>), pcr ac qug fgspgdkjcs y ghnchtrkjcs su vkacr.

;)  Nch ga vkacr fg (x>) sg prcngfg k rggjpakzkr gsg vkacr gh ak gnukniþh pkrk ghnchtrkr (x?). ?)  Nch acs vkacrgs fg (x>) y (x?) sg prcngfg k ghnchtrkr (x;) rggjpakzkhfc sus vkacrgs y fgspgdkhfc (x;).

;>

 

WHIFKF Hrc. ;

   

-?0k7k-;+70k  ? 6k x : -?k ( ;0k77k-k-;>72k )

 

6)  Fgtgrjihkr acs vkacrgs fg k  pkrk qug ga sistgjk hc tghbk scauniþh ôhink, y qug tipc fg scauniþh tighg. ‑

‚

;)  \krk ghnchtrkr acs vkacrgs pcsioags fg k  pkrk ak nuka ga sistgjk hc tighg scauniþh ôhink, sg fgog ghnchtrkr ga vkacr fg su fgtgrjihkhtg ‑

‚

?)  Whk vgz qug sg `kyk ghnchtrkfc g hnchtrkfc ga vkacr fga fgtgrjihkhtg, gstg sg `kng fiegrghtg fg < y sg ghnughtrk acs vkacrgs fg k  pkrk ak nuka ga sistgjk hc tighg scauniþh ôhink ‑

‚

>)  ^ighfc > ga vkacr fg k  pkrk ga nuka ga sistgjk si stgjk hc tighg scauniþh ôhink, sg prcngfg k rggjpakzkrë gh hugstrc sistgjk pkrk vgrieinkr qug pksk ‑

‚

;6

 

WHIFKF Hrc. ;

   

6)  Nukhfc sg rggjpakzk ga vkacr fg k  gh ga sistgjk, ga rkhbc fg ak jkrtiz cribihka gs fiegrghtg fga fg su kjpaikfk, kj paikfk, pcr ac qug sg pugfg ksujir qug ga sistgjk sist gjk hc tighg scauniþh ‑

❕ 

‚

Nukhfc k:>, ga sistgjk hc tighg scauniþh ôhink

;7