Scene 1 (0s)
g¨vwUª· Ges wbY©vqK Metrices and Determinants f~wgKv Introduction exRMvwYwZK ivwk, PjK ev c¨vivwgUvi mg~n‡K †kÖwYe×fv‡e mvwi I Kjv‡gi wfwˇZ Dfqcv‡k¦© Dj¤^ †iLv ev eÜbxi gva¨‡g cÖKvk Kivi exRMvwYwZK c×wZ Bn‡jv g¨vwUª·| AvaywbK MwY‡Z g¨vwUª· GKwU kw³kvjx †KŠkj hvi mvnv‡h¨ e¨emvq evwY‡R¨i Drcv`b, µq-weµq msµvšÍ RwUj mgm¨v AwZ mn‡RB mgvavb Kiv m¤¢e| e¨emvq evwY‡R¨i wewfbœ Z_¨vw`‡K g¨vwUª· AvKv‡i mvwR‡q Gi mvnv‡h¨ Drcv`b †gvU Avq, †gvU e¨q, jvf, ÿwZ, GKK cÖwZg~j¨ wewfbœ Pj‡Ki gvb BZ¨vw` wbiƒcb Kiv m¤¢e nq| A_©bxwZ Ges e¨emvq †ÿ‡Î e¨eüZ GK gvwÎK mgxKiY¸‡jv mgvav‡bi †ÿ‡Î g¨vwUª‡·i e¨envi I ¸iæZ¡ jÿ¨ Kiv hvq| cÖ_g James Joseph Sylvester (1814-1897) g¨vwUª‡·i aviYv †`b| wKš‘, Arthur Cayley (1821- 1895) †K g¨vwUª‡·i RbK ejv nq Ges wZwbB cÖ_‡g we‡kølYg~jKfv‡e g¨vwUª· cÖKvk K‡ib| G BDwb‡U g¨vwUª‡·i msÁv, wewfbœ cÖKvi g¨vwUª·, wecixZ g¨vwUª· wbY©q, g¨vwUª‡·i †hvM-we‡qvM, g¨vwUª‡·i ¸Y, wbY©vqK Ges g¨vwUª‡·i i¨v¼ BZ¨vw` welq¸‡jv wb‡q Av‡jvPbv Kiv n‡e| BDwbU mgvwßi mgq BDwbU mgvwßi m‡e©v”P mgq 4 w`b g~L¨ kã g¨wUª·, eM© g¨vwUª·, KY© g¨vwUª·, †¯‹jvi g¨vwUª·, A‡f` g¨vwUª·, AcÖwZmg g¨vwUª·, cÖwZmg g¨vwUª·, mgNvwZ g¨vwUª·, mn¸YK g¨vwUª·, Dj¤^ g¨vwUª·, wecixZ g¨vwUª·, A_©‡Mvbvj g¨vwUª·, mshy³ g¨vwUª·, wbY©vqK, wbY©vq‡Ki ˆewkô¨, g¨vwUª· Gi i¨v¼ †µgv‡ii wbqg BZ¨vw`| G BDwb‡Ui cvVmg~n cvV-8.1: g¨vwUª· Ges Gi cÖKvi‡f` cvV-8.2: g¨vwUª‡·i wewfbœ Kvh©µg cvV-8.3: wbY©vqK cvV-8.4: g¨vwUª‡·i i¨v¼ Ges GKNvZ wewkó mgxKi‡Yi mgvavb 8.
Scene 2 (57s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq e¨e¯'vcKxq wm×všÍ MÖn‡Y MwYZ cvV-8.1 g¨vwUª· Ges Gi cÖKvi‡f` Matrix and its types D‡Ïk¨ G cvV †k‡l Avcwb- g¨vwUª·Gi msÁv wjL‡Z cvi‡eb; wewfbœ cÖKvi g¨vwUª· Gi eY©bv Ki‡Z cvi‡eb| g¨vwUª· Matrix g¨vwUª· n‡"Q msL¨v ev cÖZxK ev exRMwYZxq ivwk‡K `yBwU eÜbxi gva¨‡g mvwi (Row) ev Kjv‡gi (Column) AvqZvKvi mvRv‡bv e¨e¯'v| g¨vwUª· e¨envi Ki‡Z mvaviYZ Z…Zxq [ ] ev cÖ_g eÜbx () A_ev cÖZxK e¨enviKiv nq| †h msL¨v ev ivwk wb‡q g¨vwUª· MwVZ nq Zv‡`i‡K g¨vwUª‡·i f~w³ (entry) ev Dcv`vb (element) ejv nq| Dcv`vb¸‡jv‡K a11, a12, a21, a22 BZ¨vw` Øviv cÖKvk Kiv nq| GQvov g¨vwUª·‡K Bs‡iRx eo Aÿi ev Capital a a 12 11 2 1 `yBwU g¨vwUª·| 5 0 a a 22 21 B = Letter Gi Øviv cÖKvk Kiv nq| †hgb, A = mvaviYfv‡e g¨vwUª· [aij] Øviv m~wPZ nq| †hLv‡b i = 1, 2, .........m mvwi msL¨v Ges j = 1, 2, ..........nKjvg msL¨v| A_©vr, A = [ aij ]mn = a a a a n 1 13 12 11 a a a a n 2 23 22 21 a a a a ........ ....... ....... n 3 33 32 31 a a a a ....... ...... ... .... .... mn m m m 3 2 1 GLv‡b g¨vwUª‡·i mvwi msL¨v m Ges Kjvg msL¨v n n‡j g¨vwUª·wUi gvÎv mn n‡e Ges co‡Z n‡e m evB n| A_©vr g¨vwUª‡·i gvÎv ev µg (Order)= mvwiKjvg| 4 3 2 5 3 1 0 4 †hgb, A = 1 2 B = GLv‡b cÖ`Ë A g¨vwUª· Gi mvwi msL¨v 2 Ges Kjvg msL¨v 2 myZivs A g¨vwUª·wUi gvÎv (Order) 22| GKBfv‡e B g¨vwUª·wUi gvÎv (Order) 23| wewfbœ cÖKv‡ii g¨vwUª· (Different types of Matrices) (i) eM©vKvi ev eM© g¨vwUª· (Square Matrix): †h g¨vwUª‡·i mvwi msL¨v I Kjvg msL¨v mgvb _v‡K ZLb Zv‡K eM© g¨vwUª· ejv nq| †hgb :22 g¨vwUª·; 33 g¨vwUª·; mn g¨vwUª·. c b a g f e B h_vµ‡g 22 Ges33 eM©vKvi g¨vwUª·| 2 1 A , 1 3 †hgb, k l h BDwbU AvU c„ôv 146.
Scene 3 (4m 18s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq g¨vwUª· Ges wbY©vqK (ii) KY© g¨vwUª· (Diagonal Matrix): hLb †Kvb eM©vKvi g¨vwUª· Gi cÖavb †KŠwYK Dcv`vbmg~n e¨ZxZ Ab¨vb¨ mKj Dcv`vbB k~Y¨ (0) _v‡K ZLb H g¨vwUª·‡K KY© g¨vwUª· ejv nq| 0 0 1 0 0 3 B C BZ¨vw` KY© g¨vwUª·| 0 1 0 0 0 0 0 1 A , 2 0 †hgb, 2 0 0 4 0 0 (iii) †¯‹jvi g¨vwUª· (Scalar Matrix): †h KY© g¨vwUª‡·i cÖavb K‡Y©i me¸‡jv Dcv`v‡bi gvb mgvb I Ak~Y¨ nq ZLb Zv‡K †¯‹jvi g¨vwUª· ejv nq| †hgb, 0 0 2 0 0 3 BZ¨vw` †¯‹jvi g¨vwUª·| a C 0 2 0 0 0 0 1 A , B , 1 0 a 0 0 2 0 0 (iv) GKK g¨vwUª· evA‡f` g¨vwUª· (Unit Matrix or Indendity Matrix): †h g¨vwUª· Gi K‡Y©i me¸‡jv Dcv`vbB GK (1) Ges Ab¨vb¨ Dcv`vb¸‡jv k~Y¨ (0) †mB g¨vwUª·‡K GKK g¨vwUª· ev Indendity Matrix ejv nq| A‡f` g¨vwUª·‡K I Øviv cÖKvk Kiv nq|mKj A‡f` g¨vwUª·B †¯‹jvi ev KY© g¨vwUª·| 0 0 1 BZ¨vw` A‡f` g¨vwUª·| 0 1 0 1 0 2I 1I , 1 0 †hgb, 1 0 0 (v) DaŸ© wÎfzR AvK…wZi g¨vwUª· (Upper Triangular Matrix): †h eM© g¨vwUª‡·i cÖavb K‡Y©i bx‡Pi mKj Dcv`vb k~Y¨ (0) nq Zv‡K EaŸ© wÎfzR AvK…wZi g¨vwUª· ejv nq| G‡ÿ‡Î aij= 0 hLb i>j a b c 4 3 2 A d e †hgb, 0 2 1 0 , B f 0 0 5 0 0 (vi) wb¤œwÎfzRAvK…wZi g¨vwUª· (Lower Triangular Matrix):†h eM© g¨vwUª‡·i cÖavb K‡Y©i Dc‡ii Dcv`vb¸‡jv k~Y¨ (0) Gi nq Zv‡K wb¤œwÎfzR AvK…wZi g¨vwUª· ejv nq| G‡ÿ‡Î aij= 0 hLb i<j. x 0 0 2 0 0 †hgb, z y 0 4 3 B 0 A , c b a 5 2 1 (vii) k~Y¨ g¨vwUª· (Zero Matrix):†h g¨vwUª· Gi mvwi I Kjv‡gi cÖwZwU Dcv`vb k~Y¨ (0) nq ZLb Zv‡K k~Y¨ 0 0 0 0 0 0 0 0 A , B 0 0 g¨vwUª· ejv nq| †hgb, 0 0 0 (vii) iƒcvšÍwiZ ev Uªv݇cvR g¨vwUª· (Transpose of Matrix): †Kvb g¨vwUª· Gi mvwi¸‡jv‡K Kjv‡g Ges Kjvg¸‡jv‡K mvwi‡Z iƒcvšÍwiZ Kiv n‡j †h g¨vwUª·wU cvIqv hv‡e, †mB g¨vwUª·‡K g~j g¨vwUª· Gi iƒcvšÍwiZ g¨vwUª· ev Transpose of Matrix ejv nq| Transpose †K [] Prime wPý Øviv cÖKvk Kiv nq| BDwbU AvU c„ôv 147.
Scene 4 (7m 16s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq e¨e¯'vcKxq wm×všÍ MÖn‡Y MwYZ a 1 b 2 B B g¨vwUª·wU n‡jv Ag¨vwUª·wUi iƒcvšÍwiZ g¨vwUª·| c 3 d c b a A 4 3 1 2 , †hgb, d 4 5 4 3 2 6 3 Avevi, B n‡j 8 7 6 0 7 4 B B g¨vwUª·wU n‡jv B g¨vwUª·wUi iƒcvšÍwiZ g¨vwUª·| 1 8 5 1 0 2 (ix) cÖwZmg g¨vwUª· (Symmetric Matrix): †h g¨vwUª‡·i mvwi‡K Kjv‡g Ges Kjvg‡K mvwi‡Z iƒcvšÍi Ki‡j Avw` g¨vwUª·wUi †Kvb cwieZ©b nq bv Zv‡K cÖwZmg g¨vwUª· ejv nq| A_©vr, GKwUeM© g¨vwUª‡·i Uªv݇cvR g¨vwUª· GKB n‡j ( A= A) Zv‡K cÖwZmg g¨vwUª· ejv nq| cÖwZmg g¨vwUª‡·i †ÿ‡Î aij = aji j e a 5 2 1 †hgb, g b e 0 3 2 A , B c g f 4 0 5 (x) wecÖwZmg ev AcÖwZmg g¨vwUª· (Skew-symmetric Matrix): †h eM© g¨vwUª· Gi Abyf~wgK mvwii Dcv`vb (Row) Ges Dj¤^ mvwii (Column) Dcv`vbmg~n GKB gv‡bi wecixZ wPý hy³ nq Ges cÖavb †KŠwYK Dcv`vbmg~‡ni gvb k~Y¨ nq Z‡e H g¨vwUª· †K wecÖwZmg ev AcÖwZmg g¨vwUª· ejv nq| e f 0 2 5 0 †hgb, A B e g 0 2 0 0 , f g 5 0 0 0 Skew symmetry matrix-Gi Gÿ‡Î aij =aji hLb ij Ges aij = 0 hLb i = j (xi) Kjvg g¨vwUª· (Column Matrix): †h g¨vwUª· Gi †KejgvÎ GKwU Kjvg _v‡K, Zv‡K Column Matrix ejv nq| †hgb, A c b a (xii) mvwi ev Abyf~wgK g¨vwUª· (Row Matrix): †h g¨vwUª· Gi mvwii msL¨v GKwU Zv‡K mvwi g¨vwUª· ev Row Matrix ejv nq| †hgbÑA = [ 1 2 3 ], B = [ abc ]. Row Matrix †K Row vector I ejv nq| (xiii) GKvZ¡‡evaK g¨vwUª· / e¨wZµgx g¨vwUª· (Singular Matrix): hw` †Kvb g¨vwUª· Gi wbY©vq‡Ki gvb k~Y¨ nq 3 2 A GLv‡b, A 0 3 2 Z‡e Zv‡K e¨wZµgx g¨vwUª· ev Singular Matrix ejv nq| †hgb, (xiv) Ae¨wZµgx g¨vwUª· (Non-singular Matrix): hw` wbY©vq‡Ki gvb k~Y¨ bv nq A_©vr, A 0 ZLb Zv‡K 2 1 A GLv‡b, A 7 1 3 Ae¨wZµgx g¨vwUª· ev Non-singular Matrix ejv nq| †hgb, (xiv) Dc g¨vwUª· (Sub Matrix): GKwU g¨vwUª‡·i †h †Kvb Dcv`vb wb‡q MwVZ Aci g¨vwUª·‡K Avw` g¨vwUª‡·i 3 2 1 Dcg¨vwUª· ejv nq| †hgb, 1 2 1 2 1 A 1 0 A , Sub Matrix of 3 1 0 BDwbU AvU c„ôv 148.
Scene 5 (11m 49s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq g¨vwUª· Ges wbY©vqK (xv) mgNvwZ g¨vwUª· (Idempotent Matrix): GKB g¨vwUª·‡K evi evi ¸Y Kivi d‡j ¸Ydj hw` Avw` g¨vwUª· nq ZLb H Matrix †K Idempotent Matrix ejv nq| A_©vr GKwU eM© g¨vwUª· A Gi Rb¨ A.A = A n‡j Zv‡K mgNvwZ g¨vwUª· ev Idempotent Matrix ejv nq| 0 0 1 4 2 2 †hgb, 0 1 0 4 3 1 A GKwU mgNvwZ g¨vwUª· A A A . Ges B GKwU mgNvwZ g¨vwUª· B B B . 3 2 1 1 0 0 (xvi) A_©‡Mvbvj g¨vwUª· (Orthogonal Matrix): hw` †Kvb g¨vwUª·‡K Zvi iƒcvšÍwiZ g¨vwUª· (A) Øviv ¸Yb Ki‡j ¸Ydj hw` GKK g¨vwUª· nq A_ev hw` †Kvb iƒcvšÍwiZ g¨vwUª· Zvi wecixZ g¨vwUª· Gi mgvb nq Z‡e Zv‡K Orthogonal Matrix ejv nq| A_©vr, AA = I A_ev A = A-1 (xvii) Dj¤^ g¨vwUª· (Vertical Matrix): †h Matrix Gi Kjvg msL¨v A‡cÿv mvwi msL¨v AwaK _v‡K, Zv‡K Dj¤^ 3 1 g¨vwUª· ejv nq| †hgb, 1 2 A 4 0 (xviii) wecixZ g¨vwUª· (Inverse Matrix): GKwUeM© g¨vwUª‡·i wecixZ g¨vwUª· cvIqv hv‡e hw` Zv Ae¨wZµgx nq A_©vr wbY©vqK k~Y¨ bv nq| Ae¨wZµgx g¨vwUª· A Gi Rb¨ hw` Ggb g¨vwUª· B cvIqv hvq †hb AB=BA=I nq Zvn‡j B g¨vwUª·wU‡K A g¨vwUª‡·i wecixZ g¨vwUª· ejv nq| A g¨vwUª· Gi wecixZ g¨vwUª·‡K A-1 Øviv m~wPZKiv nq Ges A.A-1 = A-1.A = I. msh³y g¨vwU· A A A Avevi A-1 = (Adjoint Matrix ) | | (xix) mn¸YK g¨vwUª· (Cofactor Matrix): †KvbeM© g¨vwUª‡·i cÖwZwU Dcv`v‡bi mn¸YK wbY©q K‡i Zv‡`i mgš^‡q MwVZ g¨vwUª·‡K mnM¸Y Kg¨vwUª· ejv nq| a a a A A A 13 12 11 13 12 11 †hgb, mn ¸YK g¨vwUª· A a a a A A A A 23 22 21 23 22 21 a a a A A A 33 32 31 33 32 31 (xx) mshy³ g¨vwUª· (Adjoint Matrix): †Kvb g¨vwUª‡·i cÖwZwU Dcv`vbmg~‡ni Transpose Matrix-Øviv MwVZ g¨vwUª‡·i iƒcvšÍwiZ ev Transpose Matrix †K mshy³ g¨vwUª· ev Adjoint Matrix ejv nq| Adjoint Matrix of A = Cofactor Matrix of A1 . mvims‡ÿc: g¨vwUª· n‡"Q msL¨v ev cÖZxK ev exRMwYZxq ivwk‡K `yBwU eÜbxi gva¨‡g mvwi (Row) ev Kjv‡gi (Column) AvqZvKvi mvRv‡bv e¨e¯'v| †h g¨vwUª‡·i mvwi msL¨v I Kjvg msL¨v mgvb _v‡K ZLb Zv‡K eM© g¨vwUª· ejv nq| hLb †Kvb eM©vKvi g¨vwUª· Gi cÖavb †KŠwYK Dcv`vb mg~n e¨ZxZ Ab¨vb¨ mKj Dcv`vbB k~Y¨ (0) _v‡K ZLb H g¨vwUª·‡K KY© g¨vwUª· ejv nq| BDwbU AvU c„ôv 149.
Scene 6 (16m 15s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq e¨e¯'vcKxq wm×všÍ MÖn‡Y MwYZ cvV-8.2 g¨vwUª‡·i wewfbœ Kvh©µg Operation on Matrices D‡Ïk¨ G cvV †k‡l Avcwb- g¨vwUª‡·i mgZv Kx ej‡Z cvi‡eb; g¨vwUª‡·i †hvM-we‡qvM wbY©q Ki‡Z cvi‡eb; g¨vwUª‡·i ¸Y wbY©q Ki‡Z cvi‡eb| g¨vwUª‡·i Kvh©µg Operation on Matrices g¨vwUª‡·i Kvh©µg ej‡Z cÖavbZ g¨vwUª‡·i mgZv, †hvM, we‡qvM I ¸Yb m¤úwK©Z mgm¨vi mgvavb eySvq| g¨vwUª· msµvšÍ KwZcq wbqg i‡q‡Q †m mKj wbqg¸‡jv‡K g¨vwU&ª· Kvh©µg eve¨env‡ii †gŠwjK wbqg ejv nq| g¨vwUª‡·i mgZv (Equality of Matrices): `yBwU g¨vwUª· mgvb n‡e hw` Zv‡`i AvKvi (order) GKB nq Ges mswkøó Dcv`vbmg~n mgvb nq| A_©vr A g¨vwUª· Gi mKj Dcv`b B g¨vwUª· Gi mKj Dcv`v‡bi mgvb n‡e| a a a b b b 13 12 11 13 12 11 A Ges a a a b b b B 23 22 21 23 22 21 a a a b b b 33 32 31 33 32 31 AI B `yBwU ZLbB mgvb n‡e hLb: a11 = b11, a12 = b12, a13 = b13, a21 = b21, a22 = b22, a33 = b33, a31 = b31, a32 = b32, a33 = b33 g¨vwUª‡·i †hvM-we‡qv‡Mi wbqgvejx (Laws of Addition and Subtraction of Matrices): (K) `yBwU g¨vwUª‡·i AvKvi GKB n‡j Zv‡`i g‡a¨ †hvM I we‡qvM Kiv hvq| (L) †hvMdj ev we‡qvMdj n‡"Q mswkøó Dcv`vb¸‡jvi †hvMdj ev we‡qvMdj| , 3 2 1 A 1 2 3 B 2 1 0 3 3 5 †hgb: 23 23 A I B g¨vwUª‡·i Order GKB| 4 4 4 3 2 3 1 5 0 5 4 5 3 1 2 2 3 1 B A 2 2 0 3 2 3 1 5 0 1 2 5 3 1 2 2 3 1 B A Avevi, 1 1 3 1 1 B 0 3 2 Ges 2 2 A 22 23 GLv‡b, `yBwUi AvKvi (Order) h_vµ‡g 22 Ges 23 †h‡nZz, AvKvi mgvb bq ZvB G‡`i †hvM, we‡qvM Kiv hv‡ebv| BDwbU AvU c„ôv 150.
Scene 7 (19m 12s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq g¨vwUª· Ges wbY©vqK g¨vwUª‡·i ¸Yb (Multiplication of Matrices) (K) †¯‹jvi ¸Yb (Scalar multiplication of a matrix): k †h‡Kvb GKwU aªæeK msL¨v n‡j A k. ej‡Z Ggb GKwU g¨vwUª· eySvq hv A g¨vwUª·wUi mKj Dcv`v‡bi k ¸Y| a a a ka ka ka 13 12 11 13 12 11 n‡j A a a a ka ka ka . 3 4 B 27 36 .9 B 23 22 21 23 22 21 63 72 7 8 n‡j k A a a a ka ka ka 33 32 31 33 32 31 (L) g¨vwUª‡·i mv‡_ g¨vwUª‡·i ¸Yb (Multiplication of Matrices): (i) `yBwU g¨vwUª· Gi g‡a¨ ¸Yb ZLbB m¤¢e hLb cÖ_g g¨vwUª·-Gi Kjvg msL¨v wØZxq g¨vwUª‡·i mvwii msL¨vi mgvb n‡e| Ab¨_vq `yBwU g¨vwUª‡·i ¸Yb m¤¢e bq| A g¨vwUª· Gi AvKvi (order) 23 Ges B g¨vwUª· Gi AvKvi (order) 33 n‡j, Zv‡`i ¸Ydj AB g¨vwUª· Gi AvKvi (order) n‡e 23| (ii) ¸Y Kivi mgq cÖ_g g¨vwUª· Gi 1g mvwii mv‡_ wØZxq g¨vwUª· Gi 1g Kjv‡gi Dcv`vbmg~‡ni mv‡_ ¸Y K‡i †hvM Ki‡Z n‡e| GB †hvMdj bZzb g¨vwUª· Gi 1g mvwi Ges 1g Kjv‡gi Dcv`vb n‡e| AZtci cÖ_g g¨vwUª· Gi 1g mvwi wVK †i‡L wØZxq g¨vwUª‡·i 2q Kjvg ¸Y K‡i †hvM Ki‡j cvIqv hv‡e bZzb g¨vwUª· Gi 1g mvwi 2q Kjv‡gi Dcv`vb| Abyiƒcfv‡e 3q Kjvg ¸Y K‡i †hvM Ki‡j cvIqv hv‡e bZzb g¨vwUª‡·i 1g mvwi 3q Kjv‡gi Dcv`vb| cieZx©‡Z cÖ_g g¨vwUª‡·i 2q mvwii mv‡_ Abyiƒcfv‡e wØZxq g¨vwUª‡·i 1g Kjvg, 2q Kjvg Ges 3q Kjvg ¸Y K‡i †hvM Ki‡Z n‡e| me©‡k‡l cÖ_g g¨vwUª‡·i 3q mvwii mv‡_ c~‡e©i b¨vq wØZxq g¨vwUª‡·i Kjv‡gi mv‡_ ¸b K‡i †hvM Ki‡Z n‡e| 1 B 1 nq Zvn‡j AB -Gi gvb wbY©q Kiæb| 3 1 2 3 D`vniY 1: hw` 5 2 A Ges 1 1 5 2 )1 ( 2 5 ( 3) 1 2 3 2 )1 1 ( 3 ( 3) 1 1 3 1 2 3 mgvavb: Zvn‡j, 5 2 AB 8 13 6 1 9 1 5 8 10 2 15 2 AB Ges Zvn‡j P ¸Yb Q wbY©q Kiæb| c b a P Q 4 3 2 D`vniY 2: hw` 23 z y x 31 Ges c b a P Q 4 3 2 mgvavb: †`Iqv Av‡Q, 23 z y x 31 z y x cz by ax PQ 4 3 2 21 3 2 0 A 3 6 7 B n‡j A B Gi gvb wbY©q Kiæb| 4 1 2 5 4 1 D`vniY 3: hw` Ges 3 6 7 6 8 7 3 3 6 2 7 0 3 2 0 B A 5 4 1 4 1 2 9 5 3 5 4 4 1 1 2 mgvavb: BDwbU AvU c„ôv 151.
Scene 8 (23m 22s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq e¨e¯'vcKxq wm×všÍ MÖn‡Y MwYZ 5 2 2 A nq Zvn‡j A 3 Gi gvb wbY©q Kiæb| 2 1 4 D`vniY 4: hw` 5 2 3 A 2 1 4 mgvavb: GLv‡b 4 3 12 9 Uªv݇cvR g¨vwUª· A .3 A 1 2 3 6 2 5 6 15 1 1 3 1 3 1 A Ges 4 3 2 2 4 1 B nq Zvn‡j †`Lvb †h, AB BA | D`vniY 5: hw` 6 5 4 9 6 1 1 1 3 1 3 1 = 9 2 3 6 4 9 1 1 3 = 4 7 3 mgvavb: AB 4 3 2 36 6 2 24 12 6 4 3 2 44 42 9 2 4 1 6 5 4 54 10 4 36 20 12 6 5 4 60 44 7 9 6 1 17 15 5 1 3 1 1 1 3 6 1 12 5 9 1 4 6 3 = BA 2 4 1 4 3 2 12 1 16 10 1 12 8 8 3 27 23 3 9 6 1 6 5 4 9 24 1 45 1 18 36 12 3 77 64 21 ABBA (cÖgvwYZ) 4 2 2 4 3 1 A GKwU mgNvwZ g¨vwUª· ev Idempotant matrix | D`vniY 6: †`Lvb †h, 3 2 1 mgvavb: mgNvwZ g¨vwUª· †`Lv‡Z n‡j, A2 =A.A= A cÖgvY Ki‡Z n‡e| GLb, A2 = A.A 4 2 2 4 2 2 8 12 8 8 6 4 4 2 4 4 2 2 A 4 3 1 4 3 1 12 12 4 8 9 2 4 3 2 4 3 1 3 2 1 9 8 4 6 6 2 3 2 2 3 2 1 3 2 1 g¨vwUª· A mgNvwZ g¨vwUª· ev Idempotant matrix. 1 , 1 B Ges 2 1 C nq Zvn‡j †`Lvb †h, ) ( ) ( A BC AB C . 1 3 2 A 2 0 3 D`vniY 7: hw` 2 2 3 2 3 1 3 2 AB 4 0 3 2 0 3 7 mgvavb: 2 1 1 6 3 14 7 GLb, 2 7 1 3 . AB C BDwbU AvU c„ôv 152.
Scene 9 (27m 32s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq g¨vwUª· Ges wbY©vqK 2 1 BC 2 1 Avevi, 2 1 2 1 1 4 2 2 1 4 4 6 3 2 2 1 3 2 6 3 ) .( BC A 2 1 6 0 8 4 0 3 14 7 0 2 3 GLb, 4 2 myZivs, (AB) C = A(BC) (cÖgvwYZ) , 5 1 B nq Zvn‡j 3A+5B+x=0 Gi †_‡K x gvb wbY©q Kiæb| 1 9 A 12 7 3 4 D`vniY 8: hw` mgvavb: †`Iqv Av‡Q, 3A + 5B + x = 0 x = – 3A – 5B 1 3 9 x 5 5 1 3 4 12 7 GLb, 3 27 25 5 25 3 5 27 28 32 9 12 60 35 60 9 35 12 69 47 28 32 x 69 47 2 0 2 B 0 1 1 D`vniY 9: hw` I x x f x 4 5 ) ( 2 Ges 1 0 0 Zvn‡j Gi f (B) gvb wbY©q Kiæb| mgvavb: I B B f B 4 5 ) ( 2 0 0 1 Avgiv Rvwb, I 0 1 0 1 0 0 2 0 2 2 0 2 GLb, 2 B 1 1 0 1 1 0 0 0 1 0 0 1 4 0 6 0 0 4 0 0 0 2 0 4 = 0 1 0 0 1 0 1 0 0 1 1 1 0 0 2 0 0 0 0 0 2 2 0 2 I B B f B 4 5 ) ( 2 6 0 4 2 0 2 1 0 0 1 1 1 5 0 1 1 4 0 1 0 2 0 2 1 0 0 0 0 1 BDwbU AvU c„ôv 153.
Scene 10 (30m 58s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq e¨e¯'vcKxq wm×všÍ MÖn‡Y MwYZ 4 0 6 10 0 10 0 0 4 1 1 1 5 5 0 0 4 0 2 0 2 0 0 5 4 0 0 0 10 4 0 0 0 4 10 6 6 0 0 0 5 1 4 5 1 0 0 1 6 10 1 4 0 2 0 0 0 0 5 2 6 0 3 D`vniY 10: †Kvb GKwU †`vKv‡bi wZb w`‡bi wewfbœ eªv‡Ûi Kjg weµ‡qi Z_¨ wb‡¤œi ZvwjKvq †`Lv‡bv n‡jv: Kj‡gi msL¨v w`b cvBjU B‡qv_ gb‡U· g¨vUv‡Wvi 1g w`b 3 4 10 20 2q w`b 2 3 15 20 3q w`b 1 5 12 14 cÖwZ Kj‡g jvf (UvKvq) 0.50 0.75 0.50 0.40 g¨vwUª‡·i mvnv‡h¨ †`vKv‡bi wZb w`‡bi jvf wbY©q Kiæb| 50 .0 20 10 4 3 mgvavb: 75 .0 P , 20 15 3 2 Q 50 .0 14 12 5 1 40 .0 AZGe, †gvU jvf = 𝑃 × 𝑄 50 .0 8 5 3 5.1 5. 17 20 10 4 3 75 .0 8 5.7 .2 25 1 .75 18 20 15 3 2 50 .0 6.5 6 .3 75 50 . .85 15 14 12 5 1 40 .0 mvims‡ÿc: `yBwU g¨vwUª· mgvb n‡e hw` Zv‡`i AvKvi (order) GKB nq Ges mswkøó Dcv`vbmg~n mgvb nq| A_©vr A g¨vwUª· Gi mKj Dcv`b B g¨vwUª· Gi mKj Dcv`v‡bi mgvb nq| `yBwU g¨vwUª‡·i AvKvi GKB n‡j Zv‡`i g‡a¨ †hvM I we‡qvM Kiv hvq| †hvMdj ev we‡qvMdj n‡"Q mswkøó Dcv`vb¸‡jvi †hvMdj ev we‡qvMdj| `yBwU g¨vwUª· Gi g‡a¨ ¸Yb ZLbB m¤¢e hLb cÖ_g g¨vwUª·Gi Kjvg msL¨v wØZxq g¨vwUª‡·i mvwii msL¨vi mgvb n‡e| Ab¨_vq `yBwU g¨vwUª‡·i ¸Yb m¤¢e bq| A g¨vwUª· Gi AvKvi (order) 23 Ges B g¨vwUª· Gi AvKvi (order) 33 n‡j, Zv‡`i ¸Ydj AB g¨vwUª· Gi AvKvi (order) n‡e 23| BDwbU AvU c„ôv 154.
Scene 11 (34m 40s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq g¨vwUª· Ges wbY©vqK cvV-8.3 wbY©vqK Determinant D‡Ïk¨ G cvV †k‡l Avcwb- wbY©vqK Kx Zv eY©bv Ki‡Z cvi‡eb; wbY©vq‡Ki Abyivwk Ges mn¸YK wbY©q Ki‡Z cvi‡eb; mviv‡mi wP‡Îi gva¨‡g wbY©vqK wbY©q Ki‡Z cvi‡eb; g¨vwUª· Ges wbY©vq‡Ki g‡a¨ cv_©K¨ wjL‡Z cvi‡eb| wbY©vq‡Ki aviYv Concept of Determinant wZbwU Pj‡Ki wZbwU mij mnmgxKi‡Yi mgvavb m~Î wn‡m‡e MwY‡Z wbY©vq‡Ki Avwef©ve N‡U| cieZ©x‡Z g¨vwUª‡·i aviYv I Kvh©wewa Avwef~©Z nq| wLª÷c~e© 3q kZvãx‡Z Pxb‡`kxq MwYZwe`‡`i iwPZ ÒThe Nine Chapters on the Mathematical ArtÓ eB‡Z me© cÖ_g wbY©vqK e¨eüZ nq| wbY©vqK (Determinant): wbY©vqK n‡"Q eM© g¨vwUª‡·i GKwU we‡kl cÖKv‡ii dvskb| G‡K `yBwU Dj¤^ †iLv (Vertical Line) Gi g‡a¨ †jLv nq| AeM© g¨vwUª‡·i wbY©vqK A A_ev Det(A) Øviv m~wPZ Kiv nq| b a b a A g¨vwUª‡·i mswkøó wbY©vqK n‡"Q bc ad d c d c †hgb: a a a a a a 3 2 1 3 2 1 GKBfv‡e, A g¨vwUª‡·i mswkøó wbY©vqK n‡"Q A b b b b b b 3 2 1 3 2 1 c c c c c c 3 2 1 3 2 1 D‡jøL¨ †h, cÖ‡Z¨KwU wbY©vq‡Ki GKwU gvb Av‡Q| a a a 3 2 1 A b b b wbY©vq‡Ki Abyivwk Ges mn¸YK (Minors and Cofactor of Determinant): 3 2 1 c c c 3 3 3 2 1 cÖ`Ë A wbY©vq‡Ki 9wU Dcv`vb i‡q‡Q| 9wU Dcv`v‡bi g‡a¨ †h‡Kvb GKwU Dcv`v‡bi mswkøó mvwi Ges Kjv‡gi mKj Dcv`vb ev` w`‡q Aewkó Dcv`vb¸‡jv w`‡q †h wbY©vqK nq †mwUB D³ Dcv`v‡bi mswkøó Abyivwk (Minor)| A_©vr (i,j) Zg Dcv`v‡bi Abyivwk n‡e i Zg mvwi Ges j Zg Kjv‡gi mKj Dcv`vb ev` w`‡q Aewkó Dcv`vb¸‡jv wb‡q †h wbY©vqK nq †mwU| 3 2 2 1 a a AZGe A cÖ`Ë wbY©vqKwUi (1,1)Zg Abyivwk n‡e b b , GKBfv‡e (2,3)Zg Abyivwk n‡e c c c c 3 2 2 1 3 1 a a Ges (3,2)Zg Abyivwk n‡e b b 3 1 GKBfv‡e 9wU Dcv`v‡bi g¨vwUª· †_‡K 9wU Aby ivwk cvIqv hv‡e| †h‡Kv‡bv GKwU Dcv`v‡bi Abyivwki mvg‡b h_vh_ wPý emv‡j Zv‡K H Dcv`v‡bi mn¸YK (Cofactor) ejv nq| BDwbU AvU c„ôv 155.
Scene 12 (38m 8s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq e¨e¯'vcKxq wm×všÍ MÖn‡Y MwYZ wbY©vq‡Ki we¯Í…wZ KiY (Expansion of Determinant): GKwU wbY©vqK‡K Zvi †h‡Kvb GKwU mvwi A_ev GKwU Kjv‡gi gva¨‡g we¯Í…Z Kiv hvq| wbY©vq‡Ki we¯Í…wZ n‡"Q GKwU mvwii A_ev Kjv‡gi Dcv`vb¸‡jvi mv‡_ mswkøó mnM¸Y‡Ki ¸Ydj¸‡jvi mgwó| a a a 3 2 1 A b b b 3 2 1 wbY©vq‡Ki 3 2 1 , , a a a f~w³¸‡jvi mn¸YK h_vµ‡g 3 2 1 , , A A A n‡j wbY©vq‡Ki gvb n‡e c c c 3 2 1 3 3 2 2 1 1 a A a A a A = 3 2 3 1 2 1 b a b b b a b b a + 1 c c 2 c c 3 c c 3 2 3 1 2 1 2 1 1 2 3 3 1 1 3 2 3 2 2 3 1 b c a b c b c a b c b c a b c 3 2 1 3 1 2 2 3 1 2 1 3 1 3 2 1 2 3 a b c a b c a b c a b c a b c a b c Z…Zxq gvÎvi wbY©vq‡Ki wPý = 3 2 1 D`vniY 1: A Gi wbY©vqK wYY©q Kiæb| 6 5 4 0 9 8 3 2 1 A 6 5 4 6 5 1 mgvavb: 30 66 96 12 96 54 40 3 36 48 2 0 54 1 0 9 8 6 24 0 9 5 34 0 8 0 9 8 mviv‡mi wP‡Îi gva¨‡g wbY©vq‡Ki we¯Í…wZ (Expansion of a Determinant using Sarrus Diagram) 3 a 2 a 2 a a1 a1 a a a 3 2 1 wbY©vq‡Ki Rb¨ Gi mviv‡mi wPÎ 1b 3b 1b A b b b 3 2 1 b2 b2 c c c 3 2 1 1c 3c 1c c2 c2 GKB Zxi wPý eivei Dcv`vb wZbwU ¸Y n‡e| Dci †_‡K wb‡P Zxi wPý eivei ¸Yd‡ji c~‡e© Ô+Õ wPý Ges wbP †_‡K Dc‡i ¸Yd‡ji c~‡e© ÔÑÕ wPý ewm‡q QqwU ¸Ydj †hvM Ki‡jB wbY©vqKwUi gvb cvIqv hv‡e| cÖ`Ë wbY©vqKwUi gvb n‡e 2 3 1 1 3 2 3 1 2 3 1 2 3 1 2 3 1 2 c b a c b a c b a a b c a b c a b c 3 2 1 Gi mviv‡mi wP‡Îi gva¨‡g wbY©vqK wbY©q Kiæb| A 6 5 4 D`vniY 2: 0 9 8 2 2 1 3 1 3 2 1 mgvavb: wbY©vqK Gi mviv‡mi wPÎ 4 6 4 A 6 5 4 5 5 0 9 8 8 9 9 0 8 BDwbU AvU c„ôv 156.
Scene 13 (41m 37s)
[Audio] evsjv‡`k Dš§y³ wek^we`¨vjq g¨vwUª· Ges wbY©vqK cÖ`Ë wbY©vqKwUi gvb n‡eÑ 8 5 3 9 6 1 0 4 2 3 4 9 2 6 8 1 5 0 0 54 96 108 120 0 33 207 174 wbY©vq‡Ki ˆewkó¨ (Properties of Determinants) (i) wbY©vq‡Ki mvwi ev Kjvgmg~n ci¯úi ¯'vb wewbgq Ki‡j Zvi gv‡bi †Kv‡bv cwieZ©b nq bv| c a c a b a KviY, bc ad d c d c b a Ges bc ad d b †hgb: d b (ii) wbY©vq‡Ki cvkvcvwk `yBwU mvwi ev Kjv‡gi g‡a¨ ¯'vb wewbgq Ki‡j, †m‡ÿ‡Î wbY©vq‡Ki wP‡ýi cwieZ©b N‡U, wKš' mvswL¨K gvb GKB _v‡K| 2 3 †hgb: 1 3 4 2 1 3 2 Ges 1 4 3 1 2 Kjvg `yBwU ¯'vb wewbgq Kivq gvb 1†_‡K Ñ1 n‡q‡Q| (iii) wbY©vq‡Ki `yBwU mvwi ev Kjvg Awfbœ n‡j wbY©vq‡Ki gvb k~b¨ n‡e| 6 5 †hgb: y 0 y x x Ges 0 6 5 (iv) wbY©vq‡Ki †h †Kvb mvwi ev Kjv‡gi me¸‡jv gvb k~b¨ n‡j wbY©vq‡Ki gvb k~b¨ nq| †hgb: 0 0 0 b a Ges 0 0 0 y x (v) wbY©vq‡Ki †h †Kvb GKwU mvwi ev Kjv‡gi mKj Dcv`vb‡K GKB aªæeK msL¨vØviv ¸Y Ki‡j wbY©vq‡Ki gvb‡K H aªæeK msL¨vØviv ¸Y Kiv‡K wb‡`©k K‡i| 9 2 3 2 D GLb 2q Kjvg‡K 3 Øviv ¸Y Ki‡j wbY©vqK n‡e 3D 15 9 24 12 1 †hgb: 5 4 1 (vi) †Kvb wbY©vq‡Ki †Kv‡bv mvwi ev Kjv‡gi mKj Dcv`v‡bi †hvMdjiƒ‡c cÖKvk Kiv n‡j, wbY©vqKwU‡K GKvwaK wbY©vq‡Ki †hvMdjiƒ‡c cÖKvk Kiv hvq| a a a a a a a a 3 2 1 3 2 1 3 2 †hgb: A b b b b b b b b 3 2 1 3 2 1 3 2 c c c c c c c c 3 2 1 3 2 1 3 2 (vii) wbY©vq‡Ki †Kv‡bv mvwi ev Kjv‡gi Dcv`vb¸‡jv‡K Aci GKwU mvwi ev Kjv‡gi mswkøó Dcv`vb¸‡jvi mn¸YK Øviv ¸Y Ki‡j cÖvß ¸Ydj¸‡jvi exRMwYZxq mgwó k~b¨ n‡e| a a a 3 2 1 †hgb: b b b A 3 2 1 c c c 3 2 1 wbY©vqK 0 3 3 2 2 1 1 b A b A b A †hLv‡b 3 2 1 , , A A A h_vµ‡g 3 2 1 , , a a a Gi mn¸YK| BDwbU AvU c„ôv 157.