o/u opkj Hkh# tu] ugh vkjehks dke] oi s[k NksM+s qj e/;e eu dj ';kea iq#"k lg ldyi dj] lgs oi vusd] ^uk^ u NksM+s /;s; dks] j?kqj jk[ks VsdAA jp% ekuo /kez iz.ksk l~q# Jh j.knksm+klth egkjkt STUDY PAKAGE Sujet : Mthetis Topi : DETERMINANTS & MATRIES Aville Olie : www.mthsbsuhg.o R Ide. Theo. Shot Revisio. Eeise (E. + 5. Assetio & Reso 5. Que. o opt. Es. 9 Ys. Que. o IIT-JEE(Adved 7. 5 Ys. Que. o AIEEE (JEE Mi Studet s Ne : lss Roll No. : : Addess : Plot No. 7, III- Floo, Ne Ptid Studio, Aove Bod lsses, Zoe-, M.P. NAGAR, Bhopl : 9 9 777 9, 989 5888, WhtsApp 99 559 www.tekolsses.o www.mthsbsuhg.o
. D e i itio : Let us oside the equtios +, + we epess this eliit s The sol Its vlue is give : D. E p sio o D et ei t : is lled the deteit o ode two. The sol is lled the deteit o ode thee. Its vlue e oud s: D Deteit + D +... & so o.. I this e we epd deteit i ws usig eleets o ; R, R, R o,,. M i o s : The io o give eleet o deteit is the deteit o the eleets whih ei te deletig the ow & the olu i whih the give eleet stds. Fo eple, the io o i is & the io o is. Hee deteit o ode two will hve ios & deteit o ode thee will hve 9 ios.. oto: oto o the eleet ij is ij ( i+j. M ij ; Whee i & j deotes the ow & olu i whih the ptiul eleet lies. Note tht the vlue o deteit o ode thee i tes o Mio & oto e witte s: 5. D M M + M OR D + + & so o. Tsp ose o D et e it: The tspose o deteit is deteit otied te itehgig the ows & olus. D T D. S et i, Skew-S eti, As et i D et ei t s: (i A deteit is seti i it is idetil to its tspose. Its i th ow is idetil to its i th (ii olu i.e. ij ji o ll vlues o ' A deteit is skew-seti i i it ' d is idetil ' j ' to its tspose hvig sig o eh eleet iveted i.e. ij ji o ll vlues o ' i ' d ' j '. A skew-seti deteit hs ll eleets zeo i its piipl digol. (iii A deteit is seti i it is eithe seti o skew-seti. 7. Pop eties o Det ei ts: (i The vlue o deteit eis ulteed, i the ows & olus e ite hged, i.e. D D (ii I two ows (o olus o deteit e itehged, the vlue o deteit is hged i sig ol. e.g. Let D & D The D D. NOTE : A skew-seti deteit o odd ode hs vlue zeo. Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. OR Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge o 5
Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge o 5 (iii I deteit hs ll the eleets zeo i ow o olu the its vlue is zeo, i.e. D. (iv I deteit hs two ows (o olus idetil, the its vlue is zeo, i.e. D. (v I ll the eleets o ow (o olu e ultiplied the se ue, the the deteit is ultiplied tht ue, i.e. D d D K K K The D KD (vi I eh eleet o ow (o olu e epessed s su o two tes the the deteit e epessed s the su o two deteits, i.e. z z (vii The vlue o deteit is ot lteed ddig to the eleets o ow (o olu ostt ultiple o the oespodig eleets o othe ow (o olu, i.e. D d D. The D D. Eple : Sipli Let R R + R + R ( + + Appl, ( + + ( + + (( ( ( ( + + ( + + ( + + ( + + Eple : Sipli Give deteeit is equl to Appl,
Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge o 5 ( ( ( ( [ + + + + + ] ( ( [( + + ( + + ] ( ( ( ( + + Use o to theoe. USE OF FATOR THEOREM TO FIND THE VALUE OF DETERMINANT I puttig the vlue o deteit vishes the ( is to o the deteit. Eple : Pove tht ( ( ( ( + + usig to theoe. Let D Hee ( is to o deteit Siill, let, D, D Hee, ( ( ( is to o deteit. But the give deteit is o ith ode so ( ( ( ( ( + + + µ ( + + Sie this is idetit so i ode to id the vlues o d µ. Let,, ( ( µ ( µ....(i Let,, ( ( (5 + µ 5 + µ...(ii o (i d (ii d µ Hee ( ( ( ( + +. Sel Ptie Poles. Fid the vlue o. As.. Sipli. As.. Pove tht ( + +.. Show tht ( ( ( usig to theoe. 8. M ultip li t io O Two Deteits:
Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge 5 o 5 We hve ultiplied hee ows ows ut we lso ultipl ows olus, olus ows d olus olus. I ij is deteeit o ode, the the vlue o the deteit A ij. This is lso kow s powe oto oul. Eple : Fid the vlue o d pove tht it is equl to 8. ( 8 Eple : Pove tht Give deteit e splitted ito podut o two deteits i.e. Eple : Pove tht ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Eple : Pove tht R os( Q os( P os( R os(b Q os(b P os(b R os(a Q os(a P os(a R os( Q os( P os( R os(b Q os(b P os(b R os(a Q os(a P os(a si sir ososr sisiq ososq si sip ososp sib sir osb osr sib siq osbosq sib sip osb osp si A sir os A osr si A siq os A osq si A sip os A osp si os sib osb si A os A sir siq sip osr osq osp.
Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge o 5 Sel Ptie Poles. Fid the vlue o As. (. I A, B, e el ues the id the vlue o os(b os(a B os( B os(a A os( A os(b. As. 9. Su tio o D et e its Let ( h( g( ( whee,,,,, e ostts idepedet o, the ( h( g( ( Hee utio o e the eleets o ol oe ow o olu. Noe o the eleets othe the tht ow o olu should e depedet o. I oe th oe olu o ow hve eleets depedet o the ist epd the deteit d the id the sutio. Eple : Evlute os Solutio : D os ( os Eple : D evlute D Solutio : D.........
( Eple : I, id O epsio o deteiet, we get D ( ( + 7 + + 8 + Sel Ptie Pole. Evlute D ( ( z. Itegtio o dete it ( g( h( Let ( whee,,,,, e ostts idepedet o. Hee ( d ( d g( d h( d ( + As. Note : I oe th oe ow o oe olu e utio o the ist epd the deteit d the itegte it. os / Eple : I ( os, the id ( d os Hee ( os ( os os os os os Eple : I / so os d ( d si / d. Dieetit io o D et ei t : ( ( ( Let ( g ( h ( g h ( ( g h ( ( d, the id d ( d Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge 7 o 5 Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't.
( ( ( the ( g( g( g( + h ( h ( h ( Eple : I ( ( Eple : ( g ( g ( g ( + h ( h ( h ( Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. ( (, the id the vlue o (. ( g( h ( ( g( h ( ( g( h ( ( (. Let e epeted oot o qudti equtio ( d A(, B( d ( e poloil o degee, d 5 espetivel, the show tht A( B( ( A( A ( Let g( B( B ( A( A( A ( A ( ( ( B( B( B ( B ( g( A( B( A ( B ( Sie g( g( ( ( ( ( ( ( divisile (. g( ( h( i.e. is the epeted oot o g( d h( is poloil epessio o degee. Also ( hve epeted oot. So g( is divisile (. Eple : Pove tht F depeds ol o, d Solutio : Eple : Solutio : F d sipli F. df d + Hee F is idepedet o. df df Siill. d d Hee F is idepedet o d lso. So F is depedet ol o,, Put,, F ( ( (. si I os ( Put i e os e si ( A + B + +..., the id the vlue o A d B. A + B + +... + Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge 8 o 5
A A. Dieetitig the give deteit w..t, we get e os os ( Put, we get + B + A, B Sel Ptie Pole + si si B + +... e. I + + + d. Fid (i d As. [ ] (ii + + + d As. [ 5]. (iii As. [ ] e's Rule: Sste o Lie Equtios (i Two Viles ( ( osistet Equtios: Deiite & uique solutio. [ itesetig lies ] Iosistet Equtio: No solutio. [ Pllel lie ] ( Depedet equtio: Iiite solutios. [ Idetil lies ] Let + + & + + the: Give equtios e iosistet & (ii (iii ( ( ( Give equtios e depedet Thee Viles Let, + + z d... (I + + z d... (II + + z d... (III The, D, Y D, Z D. D D D d Whee D d ; D ; D d & D d d d d osiste o sste o Equtios I D d ltest oe o D, D, D, the the give sste o equtios e osistet d hve uique o tivil solutio. I D & D D D, the the give sste o equtios e osistet d hve tivil solutio ol. I D D D D, the the give sste o equtios hve eithe iiite solutios o o solutio. (Ree Eple & Sel Ptie Pole with* (d I D ut tlest oe o D, D, D is ot zeo the the equtios e iosistet d hve o solutio. (e I give sste o lie equtios hve Ol Zeo Solutio o ll its viles the the give equtios e sid to hve TRIVIAL SOLUTION. (iv Thee equtio i two viles : I d e ot zeo, the oditio o + + ; + + & + + to e osistet i d is. Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't. d d Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge 9 o 5
Eple: Let D *Eple : Fid the tue o solutio o the give sste o equtios. + + z + + z + + 5z 5 ppl, D Now, D D 5 D R R R, R R R 5 D 5 D But D Hee o solutio Solve the ollowig sste o equtios + + z + + z + + z D D, D, D Let z t + t + t Sie oth the lies e pllel hee o vlue o d Hee thee is o solutio o the give equtio. *Eple : Solve the ollowig sste o equtios + + z + + z + + z D D, D, D All the otos o D, D, D d D e ll zeos, hee the sste will hve iiite solutios. Let z t, t t t whee t, t R. Eple : oside the ollowig sste o equtios + + z + + z + + z Fid vlues o d i suh tht sets o equtio hve (i uique solutio (ii iiite solutio (iii o solutio + + z + + z + + z D Hee o seod d thid ows e idetil hee D o. Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge o 5 Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't.
D D D I the D D D o µ (i Fo uique solutio D i.e. (ii Fo iiite solutios D D D D µ. (iii Fo o solutio D Atlest oe o D, D o D is o zeo µ. Sel Ptie Poles *. Solve the ollowig sste o equtios + + z + + z + + 5z As. + t t z t whee t R. Solve the ollowig sste o equtios + + z + + z z As.,, z. Solve: ( + ( + z, ( + (z +, ( + ( + z whee + +. As.,, z. Let + + ; + 5 +, + + 5 + 8 + + + t, i the sste o equtios i d e osistet the id the vlue o t. As. t 7. Ap pli tio o Det eit s: Followig eples o shot hd witig lge epessios e: (i Ae o tigle whose veties e (, ;,, is: (ii D I D the the thee poits e ollie. Equtio o stight lie pssig though (, & (, is (iii The lies: + +... ( + +... ( + +... ( (iv e ouet i,. oditio o the osiste o thee siulteous lie equtios i viles. ² + h + ² + g + + epesets pi o stight lies i: h g + gh ² g² h² h g Teko lsses, Mths : Suhg R. Ki (S. R. K. Si, Bhopl, Phoe : 9 9 777 9, 989 5888 pge o 5 Suessul People Reple the wods like; "wish", "t" & "should" with "I Will". Ieetive People do't.