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1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 261 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 286 39 "Laboratoire 7 - M\351th ode de Gauss-Jordan" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 52 "Par Clau de St-Hilaire, claude.sthilaire@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT 310 51 "Principales commandes utilis\351es dans ce laboratoire " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "matrix de la biblioth\350que l inalg, gaussjord, augment, rref" }}{PARA 0 "" 0 "" {TEXT -1 69 "Note : S\351lectionner un mot et utiliser l'aide pour plus d'information" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT 303 7 "Rappels" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Pour r\351soudre un syst\350me d'\351quations Ax = b, l" }{TEXT 305 32 "a m\351thode d'\351limination de Gauss" }{TEXT -1 12 " transforme " }{TEXT 304 28 "la matrice augment\351e [A | b]" } {TEXT -1 68 ", en appliquant les 3 op\351rations \351l\351mentaires su r les lignes, en une" }{TEXT 306 20 " matrice \351chelonn\351e " } {TEXT -1 35 "[H | c] ayant les m\352mes solutions. " }}{PARA 0 "" 0 " " {TEXT -1 118 "Le premier nombre non nul de chaque ligne de la matric e H est appel\351 pivot et sous un pivot, l'on n'a que des z\351ros. \+ " }}{PARA 0 "" 0 "" {TEXT -1 173 "La m\351thode de Gauss-Jordan consis te, avec les op\351rations \351l\351mentaires, \340 mettre des z\351ro s au dessus et en dessous des pivots et \340 rendre les pivots \351gau x \340 1. On obtient la " }{TEXT 309 26 "matrice \351chelonn\351e r \351duite" }{TEXT -1 28 " de [A | b], qui est unique." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Si A est une matrice carr\351e (nxn), in versible (non singuli\350re), On obtient alors la solution de A x = b \+ qui est [ I | solutionx]. " }}{PARA 0 "" 0 "" {TEXT -1 23 "On sch\351 matise avec : " }{TEXT 307 29 "[ A | b] ~ [ I | solutionx ] " }} {PARA 0 "" 0 "" {TEXT -1 78 "Comme la solution de A x = b est x = A^( -1) b, on peut aussi sch\351matiser par " }{TEXT 308 28 "[ A | b] ~ [ \+ I | A^(-1) b ] " }}{PARA 0 "" 0 "" {TEXT -1 12 "Avec Maple, " }{TEXT 320 12 "gaussjord(M)" }{TEXT -1 20 " o\371 M est la matrice" }{TEXT 322 1 " " }{TEXT 319 7 "[A | b]" }{TEXT 323 1 " " }{TEXT 321 3 "ou " } {TEXT 324 9 "rref (M) " }{TEXT -1 61 "(reduce row echelon form) donne \+ la matrice \351chelonn\351e-r\351duite" }}{PARA 0 "" 0 "" {TEXT -1 110 "Voir le laboratoire 7 Gauss\311tapes pour appliquer ces op\351rat ions \351l\351mentaires, \351tape par \351tape, comme \340 la main" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 73 "Gauss-Jordan et r\351solutions simultan\351es de syst\350mes d'\351quations lin\351aires" }}{EXCHG {PARA 0 "" 0 "" {TEXT 287 10 "Exemple 1)" }{TEXT -1 60 " R\351soudre a vec Gauss-Jordan, le syst\350me d'\351quations suivant:" }}{PARA 0 "" 0 "" {TEXT -1 16 "4x - 5y + 4z = 7" }}{PARA 0 "" 0 "" {TEXT -1 35 "-2x + 8y + z = 12\n5x + 8y - 9z = 45" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M:=matrix(3,4,[4,-5,4,7,-2,8,1,12,5,8,-9,45]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gaussjord(M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "On a la solution x = " }{XPPEDIT 18 0 "23 81/479;" "6#*&\"%\"Q#\"\"\"\"$z%!\"\"" }{TEXT -1 6 ", y = " }{XPPEDIT 18 0 "1303/479;" "6#*&\"%.8\"\"\"\"$z%!\"\"" }{TEXT -1 6 ", z = " } {XPPEDIT 18 0 "86/479;" "6#*&\"#')\"\"\"\"$z%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 10 "Exemple 2)" }{TEXT -1 92 " R\351solvons maintena nt le syst\350me d'\351quations suivant qui a la m\352me matrice des c oefficients " }}{PARA 0 "" 0 "" {TEXT -1 16 "4x - 5y + 4z = 5" }} {PARA 0 "" 0 "" {TEXT -1 35 "-2x + 8y + z = 10\n5x + 8y - 9z = 15" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "M2:=matrix(3,4,[4,-5,4,5,-2, 8,1,10,5,8,-9,15]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "GJ:= rref(M2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "On a la solution x = " }{XPPEDIT 18 0 "1085/479;" "6#*&\" %&3\"\"\"\"\"$z%!\"\"" }{TEXT -1 6 ", y = " }{XPPEDIT 18 0 "805/479;" "6#*&\"$0)\"\"\"\"$z%!\"\"" }{TEXT -1 6 ", z = " }{XPPEDIT 18 0 "520/4 79;" "6#*&\"$?&\"\"\"\"$z%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Dans les 2 cas, on doit faire les m\352mes op\351rations pour r \351duire la matrice des coefficients \340 la matrice identit\351. On \+ peut faire les 2 \351tapes en une seule (" }{TEXT 257 13 "simultan\351 ment" }{TEXT -1 103 ") en partant avec la matrice des coefficients aug ment\351e des 2 colonnes des constantes(\340 droite des =)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " M3:=augment(M,[5,10,15]);# solution simultan\351e des 2 syst\350mes d' \351quations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gaussjord(M 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 204 "La quatri\350me colonne donne la solution du prem ier syst\350me d'\351quations, x = 2381/479, y = 1303/479, z = 86/479 \+ et la cinqui\350me, celle du deuxi\350me syst\350me d'\351quations, x \+ = 1085/479, y = 805/479, z = 520/479" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 311 9 "Attention" }{TEXT -1 112 " : Ici, ne pas faire backsub car cela exprimerait la cinqui\350me colonne comme combinaison lin \351aire de 4 premi\350res" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 103 " R\351solution simultan\351e de syst\350mes d'\351quations Ax = bi o \371 A est une matrice inversible (non singuli\350re)." }}{PARA 0 "" 0 "" {TEXT -1 51 "La solution de A x = b par Gauss-Jordan : [ A | b] \+ " }{TEXT 300 1 "~" }{TEXT -1 19 " [ I | solutionx ] " }}{PARA 0 "" 0 " " {TEXT -1 118 "Si on a plusieurs syst\350mes d'\351quations Ax = b1, \+ Ax = b2, Ax = b3,...Ax = bk, on r\351sout simultan\351ment avec Gauss- Jordan" }}{PARA 0 "" 0 "" {TEXT 301 83 "[ A | b1, b2 ,b3 ..,bk ] ~ [ I | solution1, solution2, solution3, ...solutionk ]" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 1 "I" }{TEXT 270 40 "nverser une matrice avec Gauss-Jor dan : " }}{EXCHG {PARA 257 "" 0 "" {TEXT 312 108 "Si A est une matrice non singuli\350re, alors on peut trouver son inverse A^(-1) avec Gaus s-Jordan appliqu\351e \340 " }{TEXT -1 10 "[ A | I ] " }{TEXT 314 37 " et on obtient la matrice \351quivalente " }{TEXT -1 13 "[ I | A^(-1)] " }{TEXT 315 0 "" }{TEXT 316 1 ":" }}{PARA 257 "" 0 "" {TEXT 313 1 " \+ " }{TEXT -1 10 "[ A | I ] " }{TEXT 260 1 "~" }{TEXT -1 14 " [ I | A^(- 1)]" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 285 11 "Explication" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 299 14 "Cas g\351n\351ral : " }}{PARA 0 "" 0 " " {TEXT -1 115 "Trouver l'inverse d'une matrice A non singuli\350re re vient \340 r\351soudre un syst\350me d'\351quations lin\351aires simul tan\351ment." }}{PARA 0 "" 0 "" {TEXT -1 182 "En effet, trouver l'inve rse d'une matrice A d'ordre n, revient \340 chercher une matrice B = [ b1,b2,b3,...bn], (o\371 les bi sont les colonnes de B) telle que AB = \+ I = [e1,e2,e3,...en].(1)" }}{PARA 0 "" 0 "" {TEXT -1 8 "Or AB= A" } {TEXT 261 0 "" }{TEXT -1 53 "[b1,b2,b3,...bn] = [Ab1, A,b2, Ab3,...Abn ] (2) " }}{PARA 0 "" 0 "" {TEXT -1 35 "Donc, trouver B revient \+ \340 r\351soudre " }}{PARA 0 "" 0 "" {TEXT -1 59 "[Ab1 A,b2, Ab3,...Ab n] = [e1,e2,e3,...en], de (1) et (2)" }}{PARA 0 "" 0 "" {TEXT -1 49 "c'est-\340-dire r\351soudre les syst\350mes d'\351quations :" }} {PARA 0 "" 0 "" {TEXT 265 44 "Ab1 = e1; Ab2 = e2; Ab3 = e3; ...Abn = e n " }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 69 "Ces syst \350mes d'\351quations ont tous la m\352me matrice des coefficients A \+ " }}{PARA 0 "" 0 "" {TEXT -1 79 "En r\351solvant Ab1 = e1 avec Gauss-J ordan, on a : [A | e1]-> [I | b1]. De m\352me, " }}{PARA 0 "" 0 "" {TEXT -1 72 "en r\351solvant Ab2 = e2 avec Gauss-Jordan, on a : [A | \+ e2]-> [I | b2]...," }}{PARA 0 "" 0 "" {TEXT -1 4 "... " }}{PARA 0 "" 0 "" {TEXT -1 69 "en r\351solvant Abn = en avec Gauss-Jordan, on a : \+ [A | en]-> [I | bn]." }}{PARA 0 "" 0 "" {TEXT 263 46 "En r\351solvant \+ simultan\351ment avec gauss -Jordan," }{TEXT -1 8 " on a : " }}{PARA 0 "" 0 "" {TEXT 264 50 "[ A | e1 e2 e3 ...en ] ~ [I | b1, b2, b3, ... bn]," }{TEXT -1 14 " c'est-\340-dire " }}{PARA 0 "" 0 "" {TEXT -1 11 " [ A | I ] " }{TEXT 268 1 "~" }{TEXT -1 18 " [I | B], et donc " } {TEXT 262 11 "[ A | I ] " }{TEXT 266 1 "~" }{TEXT 267 14 " [ I | A^(- 1)]" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 261 "" 0 "" {TEXT 289 82 "Explicitons ce qui se passe en analysant un cas particulier avec une matrice 3x3 :" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Trouver l'inverse de la matrice \+ A revient trouver la matrice B telle que AB = I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "A:=matrix(3,3,[1,10,25,2,30,35,3,40,55]):B:= matrix(3,3,[x,u,r,y,v,s,z,w,t]):Id:=diag(1,1,1):'A'=evalm(A),'B'=evalm (B),'Id'=evalm(Id);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "EquationMatricielle:=multiply(A,B)= evalm(Id);# AB = I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "De cette \+ \351galit\351 de matrices, on obtient les 3 syst\350mes d'\351quations :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "x+10*y +25*z;" "6#,(%\"xG\"\"\"*&\"#5F%%\"yGF%F%*&\"#DF%%\"zGF%F%" }{TEXT -1 18 " = 1, " }{XPPEDIT 18 0 "u+10*v+25*w;" "6#,(%\"uG\"\"\" *&\"#5F%%\"vGF%F%*&\"#DF%%\"wGF%F%" }{TEXT -1 17 " = 0, " } {XPPEDIT 18 0 "r+10*s+25*t;" "6#,(%\"rG\"\"\"*&\"#5F%%\"sGF%F%*&\"#DF% %\"tGF%F%" }{TEXT -1 6 " = 0\n " }{XPPEDIT 18 0 "2*x+30*y+35*z;" "6#,( *&\"\"#\"\"\"%\"xGF&F&*&\"#IF&%\"yGF&F&*&\"#NF&%\"zGF&F&" }{TEXT -1 15 " = 0, " }{XPPEDIT 18 0 "2*u+30*v+35*w;" "6#,(*&\"\"#\"\" \"%\"uGF&F&*&\"#IF&%\"vGF&F&*&\"#NF&%\"wGF&F&" }{TEXT -1 15 " = 1 , \+ " }{XPPEDIT 18 0 "2*r+30*s+35*t;" "6#,(*&\"\"#\"\"\"%\"rGF&F&*& \"#IF&%\"sGF&F&*&\"#NF&%\"tGF&F&" }{TEXT -1 7 " = 0 \n " }{XPPEDIT 18 0 "3*x+40*y+55*z;" "6#,(*&\"\"$\"\"\"%\"xGF&F&*&\"#SF&%\"yGF&F&*&\"#bF &%\"zGF&F&" }{TEXT -1 15 " = 0 , " }{XPPEDIT 18 0 "3*u+40*v+55 *w;" "6#,(*&\"\"$\"\"\"%\"uGF&F&*&\"#SF&%\"vGF&F&*&\"#bF&%\"wGF&F&" } {TEXT -1 15 " = 0 , " }{XPPEDIT 18 0 "3*r+40*s+55*t;" "6#,(*& \"\"$\"\"\"%\"rGF&F&*&\"#SF&%\"sGF&F&*&\"#bF&%\"tGF&F&" }{TEXT -1 4 " \+ = 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Remarque : Le premier syst \350me d'\351quations en x, y, z est quivalent " }{TEXT 327 1 " " } {TEXT 325 1 "\340" }{TEXT 328 9 " Ab1 = e1" }{TEXT -1 260 " o\371 b1 e st la premi\350re colonne de B et e1, la premi\350re colonne de la mat rice identit\351e. De m\352me, pour le deuxi\350me syst\350me d'\351qu ation en u, v, w, on Ab2 = e2 et pour le troisi\350me en r, s, t : Ab3 = e3. Donc r\351soudre les 3 syst\350mes d'\351quations revient r \351soudre " }{TEXT 326 31 "Ab1 = e1, Ab2 = e2 et Ab3 = e3." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Les 3 syst\350mes de 3 \+ \351quations 3 inconnues, ont la " }{TEXT 295 4 "m\352me" }{TEXT -1 1 " " }{TEXT 296 40 "matrice A comme matrice des coefficients" }{TEXT -1 28 ", on peut donc les r\351soudre " }{TEXT 294 13 "simultan\351men t" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "augmen t(A,[1,0,0],[0,1,0],[0,0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "GJ:=gaussjord(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 389 "La qu atri\350me colonne de GJ donne la solution du premier syst\350me : x = -5, y = 1/10, z = 1/5. Cela donne la premi\350re colonne de B\nLa cin qui\350me colonne de GJ donne la solution du deuxi\350me syst\350me : \+ u = -9, v = 2/5 , w = 1/5. Cela donne la deuxi\350me colonne de B\nLa \+ sixi\350me colonne de GJ donne la solution du troisi\350me syst\350me \+ : r = 8, s = -3/10, t = -1/5. Cela donne la troisi\350me colonne de B " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "La matrice B est donc form\351e des colonnes 4 \340 6 de la matric e GJ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(B) = augment( col(GJ,4..6));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "V\351rifions av ec la commande inverse(A) qui donne la matrice inverse de A" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 " \300 partir de l'\351quation matricielle , EquationMatricielle, on aurait pu r\351soudre avec solve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "with(student):equate(lhs(EquationMa tricielle),rhs(EquationMatricielle));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(%,\{x,y,z,u,v,w,r,s,t\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(%,op(B));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Remarque : Le premier syst\350me d'\351quations en x, y, \+ z est quivalent " }{TEXT 297 1 " " }{TEXT 290 1 "\340" }{TEXT 298 9 " \+ Ab1 = e1" }{TEXT -1 260 " o\371 b1 est la premi\350re colonne de B et \+ e1, la premi\350re colonne de la matrice identit\351e. De m\352me, pou r le deuxi\350me syst\350me d'\351quation en u, v, w, on Ab2 = e2 et p our le troisi\350me en r, s, t : Ab3 = e3. Donc r\351soudre les 3 syst \350mes d'\351quations revient r\351soudre " }{TEXT 291 31 "Ab1 = e1, Ab2 = e2 et Ab3 = e3 " }{TEXT 292 73 "qu'on r\351soud simultan\351men t.\n\nOn aurait pu aboutir au m\352me r\351sultat avec :" }{TEXT -1 128 "\nAB = A[b1,b2,b3] = [Ab1,Ab2,Ab3] et la matrice identit\351 I = \+ [e1,e2,e3] alors AB = I devient \n[Ab1,Ab2,Ab3] = [e1,e2,e3] d'o\371 : " }{TEXT 293 30 "Ab1 = e1, Ab2 = e2 et Ab3 = e3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 10 "Exemple 3)" }{TEXT -1 70 " Trouver la ma trice inverse de la matrice A suivante avec Gauss-Jordan" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=matrix(3,3,[23,-4,56,2,0,45,12,7 ,55]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Id:=matrix(3,3,[1 ,0,0,0,1,0,0,0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AI:= augment(A,Id);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gaussjord (AI);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "L'inverse de A est donc \+ :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "augment(col(%,4..6)); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "On peut appliquer Gauss-Jorda n " }{TEXT 259 15 "\351tape par \351tape" }{TEXT -1 52 " comme \340 la main : voir le laboratoire 8 GaussEtapes" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "V\351rifions en multipliant notre r\351ponse par A ou ave c la commande Maple: inverse(A) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "multiply(%,A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 271 15 "G\351n\351ralisons : " }{TEXT 282 8 "[A | B] " }{TEXT 283 1 "~" }{TEXT 284 16 "[ I | A^(-1 )*B] " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Soit A(nxn), une matrice \+ inversible. En appliquant GaussJord, on a le sch\351ma " }{TEXT 279 8 "[A | B] " }{TEXT 280 1 "~" }{TEXT 281 16 "[ I | A^(-1)*B] " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Soit B(nx k) = [b1,b2,b3,..bk]. En appliquant gauss-Jordan \340 [A | B], on ob tient : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 33 "[A | B] = [ A | b1 b2 ,b3 ..bk ] " }{TEXT 274 1 "~" }{TEXT -1 50 " [ I | A^(-1) b1, A^(-1) b2, A^(-1) b3, A(-1) bk] " }}{PARA 0 "" 0 "" {TEXT -1 86 "mais [A^(-1) b1 , A^(-1) b2, A^(-1) b3, A(-1) vk] = A^(-1)[b1,b2,b3...bk] = A^(-1)* B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "donc en appliquant GaussJord on a le sch \351ma " }{TEXT 272 8 "[A | B] " }{TEXT 275 1 "~" }{TEXT 276 16 "[ I \+ | A^(-1)*B] " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Cas particulier : Si B est la matrice identit\351 I, on a le sc h\351ma : " }{TEXT 273 9 "[A | I ] " }{TEXT 277 1 "~" }{TEXT 278 14 " \+ [ I | A^(-1)]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 1 {PARA 3 " " 0 "" {TEXT 269 9 "Exercices" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 329 5 "No 1)" }{TEXT -1 59 " R\351soudre simultan\351ment les syst\350mes d' \351quations lin\351aires:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Syst\350me 1: 5x - 7y =11 Syst\350me 2: \+ 5x - 7y = 100 Syst\350me 3: 5x - 7y = 123" }}{PARA 0 "" 0 "" {TEXT -1 110 " 12x + y = -15 12 x + y = 45 12x + y = 22 " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 317 32 " Espace de travail de l'\351tudiant" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT 330 5 "No 2)" }{TEXT -1 130 " Trouver, av ec la m\351thode de Gauss-Jordan, la matrice inverse de la matrice B e t v\351rifier votre r\351ponse avec inverse(B); suivante:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "B:=matrix(4,4,[12,4,1,-5,2,45,1,-34 ,55,2,-8,-34,5,6,12,8]);" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 318 32 " Es pace de travail de l'\351tudiant" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }