{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 296 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 301 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Tim es" 1 14 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 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 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 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 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" -1 259 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 80 "Travail 2 - Laboratoire 4 - Preuves par r\351currence (induction) avec des matrices" }{TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 52 "Par Claude St-Hilaire , claude.sthilaire@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 8 "Exemples" }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 10 "Exemple 1)" }{TEXT -1 11 " Soit A = " }{XPPEDIT 18 0 "matrix([[1, 0, -1], [0, 3, 0], [-1 , 0, 1]]);" "6#-%'matrixG6#7%7%\"\"\"\"\"!,$F(!\"\"7%F)\"\"$F)7%,$F(F+ F)F(" }{TEXT -1 4 " . T" }{TEXT 267 84 "rouver une formule pour A^n o \371 n est un nombre naturel et prouvez-la par r\351currence " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A:= matrix([[1, 0, -1], [0, \+ 5, 0], [-1, 0, 1]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "A^2 =evalm(A^2),A^3=evalm(A^3),A^4=evalm(A^4),A^5=evalm(A^5),A^6=evalm(A^6 ),A^7=evalm(A^7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "Les \351l \351ments des coins A[1,1], A[1,3], A[3,1], A[3,3] sont multipli\351s \+ par 2 \340 chaque \351tape. Dans A^n, l'\351l\351ment d'un coin sera m ultipli\351 par 2^(n-1) Exemple dans A^5, ils sont multipli\351s par 1 6 = 2^4. De plus l'\351l\351ment A[2,2] = 5^n dans A^n. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Montrons que pout tout nombre naturel n>= 1, A^n = " }{XPPEDIT 18 0 "matrix([[2^(n-1), 0, -2^(n-1)], [0, 5^n, 0] , [-2^(n-1), 0, 2^(n-1)]]);" "6#-%'matrixG6#7%7%)\"\"#,&%\"nG\"\"\"F,! \"\"\"\"!,$)F),&F+F,F,F-F-7%F.)\"\"&F+F.7%,$)F),&F+F,F,F-F-F.)F),&F+F, F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "D\351monstration : \na) le th\351or\350me est vra i pour n = 1" }}{PARA 0 "" 0 "" {TEXT -1 107 "b) Supposons que le th \351or\350me est vrai pour n = k et d\351montrons qu'il est vrai pour \+ n = k+1, c'est-\340-dire :" }}{PARA 0 "" 0 "" {TEXT -1 15 "Supposons \+ que " }{XPPEDIT 18 0 "A^k;" "6#)%\"AG%\"kG" }{TEXT -1 3 "= " } {XPPEDIT 18 0 "matrix([[2^(k-1), 0, -2^(k-1)], [0, 5^k, 0], [-2^(k-1), 0, 2^(k-1)]]);" "6#-%'matrixG6#7%7%)\"\"#,&%\"kG\"\"\"F,!\"\"\"\"!,$) F),&F+F,F,F-F-7%F.)\"\"&F+F.7%,$)F),&F+F,F,F-F-F.)F),&F+F,F,F-" } {TEXT -1 25 " (1) alors montrons que " }{XPPEDIT 18 0 "A^(k+1);" "6#) %\"AG,&%\"kG\"\"\"F'F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[2^k , 0, -2^k], [0, 5^(k+1), 0], [-2^k, 0, 2^k]]);" "6#-%'matrixG6#7%7%)\" \"#%\"kG\"\"!,$)F)F*!\"\"7%F+)\"\"&,&F*\"\"\"F3F3F+7%,$)F)F*F.F+)F)F* " }{TEXT -1 4 " (2)" }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }{XPPEDIT 18 0 "A^(k+1);" "6#)%\"AG,&%\"kG\"\"\"F'F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A^k;" "6#)%\"AG%\"kG" }{TEXT -1 0 "" }{XPPEDIT 18 0 "A;" "6#%\"A G" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[2^(k-1), 0, -2^(k-1)], [ 0, 5^k, 0], [-2^(k-1), 0, 2^(k-1)]]);" "6#-%'matrixG6#7%7%)\"\"#,&%\"k G\"\"\"F,!\"\"\"\"!,$)F),&F+F,F,F-F-7%F.)\"\"&F+F.7%,$)F),&F+F,F,F-F-F .)F),&F+F,F,F-" }{XPPEDIT 18 0 "matrix([[1, 0, -1], [0, 5, 0], [-1, 0, 1]]);" "6#-%'matrixG6#7%7%\"\"\"\"\"!,$F(!\"\"7%F)\"\"&F)7%,$F(F+F)F( " }{TEXT -1 39 " , par l'hypoth\350se (1) de l'induction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 53 "Calculons le me mbre de droite de l'\351galit\351 avec Maple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "M:=matrix([[2^(k-1), 0, -2^(k-1)], [0, 5^k, 0], \+ [-2^(k-1), 0, 2^(k-1)]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "MA:=simplify(multiply(M,A));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "On a bien le r\351sultat (2)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "donc, Si A = " }{XPPEDIT 18 0 "matrix([[1, 0, -1], [0, 5, 0], [-1, 0, 1]]);" "6#-%'matrixG6#7%7%\"\"\"\"\"!,$F(!\"\"7%F)\"\"&F)7%,$F(F+F )F(" }{TEXT -1 53 " alors, pour tout nombre naturel n>=1, on a : A^n \+ = " }{XPPEDIT 18 0 "matrix([[2^(n-1), 0, -2^(n-1)], [0, 5^n, 0], [-2^( n-1), 0, 2^(n-1)]])" "6#-%'matrixG6#7%7%)\"\"#,&%\"nG\"\"\"F,!\"\"\"\" !,$)F),&F+F,F,F-F-7%F.)\"\"&F+F.7%,$)F),&F+F,F,F-F-F.)F),&F+F,F,F-" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A:=matrix(4,4,[0,0,1,1,0,0,1 ,1,1,1,0,0,1,1,0,0]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 265 97 "Exemp le 2) Trouver une formule pour A^n o\371 n est un nombre naturel et pr ouvez-la par r\351currence :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "E xplorons d'abord en calculant : A^2," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A2:=evalm(A^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "On remarque la position des \351l\351ments a(i,j) non nuls dans A \+ et A^2. " }}{PARA 0 "" 0 "" {TEXT -1 132 "La valeur des \351l\351ments a(i,j) non nuls est 2 pour A^2. Sera-t-elle de 3 pour A^3 et quelles positions seront les a(i,j) non nuls?: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A3:=evalm(A^3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Les positions des a(i,j) non nuls pour n = 3 sont aux m\352mes po sitions pour n = 1." }}{PARA 0 "" 0 "" {TEXT -1 81 "De plus les a(i,j) non nuls valent 2 pour n = 2 et a(i,j) = 4 =2^2 pour n = 3 . " }} {PARA 0 "" 0 "" {TEXT -1 109 "La valeur des a(i,j) non nuls semble \+ \352tre de la forme 2^(n-1),. Est-ce valable pour d'autres valeurs de \+ n ?. " }}{PARA 0 "" 0 "" {TEXT -1 22 "Essayons n = 4, n =5, " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A4:=evalm(A^4);A5:=evalm(A^5 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "La position des a(i,j) non nuls est la m\352me pour les n pairs et est diff\351rente de la posit ion des a(i,j) non nuls quand n est impair. " }}{PARA 0 "" 0 "" {TEXT -1 39 "On a obtenu pour les a(i,j) non nuls :" }}{PARA 0 "" 0 "" {TEXT -1 30 "pour A^2, a(i,j) = 2^1 = 2, " }}{PARA 0 "" 0 "" {TEXT -1 30 "pour A^3, a(i,j) = 2^2 = 4, " }}{PARA 0 "" 0 "" {TEXT -1 30 " pour A^4, a(i,j) = 2^3 = 8, " }}{PARA 0 "" 0 "" {TEXT -1 29 "pour A^ 5, a(i,j) = 2^4 = 16. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Il semble donc que pour A^n, la formule des a(i,j) \+ non nuls sera de 2^(n-1)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "V \351rifions notre hypoth\350se (conjoncture) avec n = 10 et n = 11" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A10:=evalm(A^10);A11:=evalm (A^11);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "C'est conforme \340 n otre hypoth\350se : pour A^10, a(i,j) = 2^9 =512 et pour A^11, a(i,j ) = 2^10 = 1024, donc :m Postulons l'hypoth\350se :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Si " }{TEXT 263 14 " n est impair " }{TEXT -1 11 "alors A^n =" }{XPPEDIT 18 0 "matrix([[0, 0, 2^(n-1), 2^(n-1)], [0, 0, 2^(n-1), 2^(n-1)], [2^( n-1), 2^(n-1), 0, 0], [2^(n-1), 2^(n-1), 0, 0]]);" "6#-%'matrixG6#7&7& \"\"!F()\"\"#,&%\"nG\"\"\"F-!\"\")F*,&F,F-F-F.7&F(F()F*,&F,F-F-F.)F*,& F,F-F-F.7&)F*,&F,F-F-F.)F*,&F,F-F-F.F(F(7&)F*,&F,F-F-F.)F*,&F,F-F-F.F( F(" }{TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 5 "et si" }{TEXT 264 12 " n est pair " }{TEXT -1 13 "alors A^n = " }{XPPEDIT 18 0 "matrix( [[2^(n-1), 2^(n-1), 0, 0], [2^(n-1), 2^(n-1), 0, 0], [0, 0, 2^(n-1), 2 ^(n-1)], [0, 0, 2^(n-1), 2^(n-1)]]);" "6#-%'matrixG6#7&7&)\"\"#,&%\"nG \"\"\"F,!\"\")F),&F+F,F,F-\"\"!F07&)F),&F+F,F,F-)F),&F+F,F,F-F0F07&F0F 0)F),&F+F,F,F-)F),&F+F,F,F-7&F0F0)F),&F+F,F,F-)F),&F+F,F,F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "On a 2 cas \340 consid\351rer: Si l'exposant de A est i mpair, il est de la forme 2n+1 et s'il est pair, il est de la forme 2 n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "A) " }{TEXT 261 66 "Si l'exposant de A est impair, A^(2n+1) sera la matrice suivante :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "A^(2*n+1)=matrix(4,4,[0,0,2^(2*n),2^(2*n),0,0,2^( 2*n),2^(2*n),2^(2*n),2^(2*n),0,0,2^(2*n),2^(2*n),0,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "B) " } {TEXT 259 62 "Si l'exposant de A est pair, A^(2n) sera la matrice suiv ante :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "A^(2*n)=matrix(4,4,[2^(2*n-1),2^(2*n-1),0,0,2^(2*n-1 ),2^(2*n-1),0,0,0,0,2^(2*n-1),2^(2*n-1),0,0,2^(2*n-1),2^(2*n-1)]);" }} }{EXCHG {PARA 257 "" 0 "" {TEXT -1 42 "Prouvons A) par r\351currence, \+ c'est-\340-dire :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Prouvons que pour tout nombre naturel n, " }{XPPEDIT 18 0 "A^(2*n+1);" "6#)%\"AG, &*&\"\"#\"\"\"%\"nGF(F(F(F(" }{TEXT -1 2 "= " }{XPPEDIT 18 0 "matrix([ [0, 0, 2^(2*n), 2^(2*n)], [0, 0, 2^(2*n), 2^(2*n)], [2^(2*n), 2^(2*n), 0, 0], [2^(2*n), 2^(2*n), 0, 0]]);" "6#-%'matrixG6#7&7&\"\"!F()\"\"#* &F*\"\"\"%\"nGF,)F**&F*F,F-F,7&F(F()F**&F*F,F-F,)F**&F*F,F-F,7&)F**&F* F,F-F,)F**&F*F,F-F,F(F(7&)F**&F*F,F-F,)F**&F*F,F-F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "a) Le th\351or\350me est vrai pour n = 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "b) Supposons que le th\351or\350me est vrai pour n \+ = k et d\351montrons qu'il est vrai pour n = k+1 (alors 2n+1 = 2(k+1)+ 1 = 2k+3) :c'est-\340-dire :\nSupposons que" }{XPPEDIT 18 0 "A^(2*k+1 );" "6#)%\"AG,&*&\"\"#\"\"\"%\"kGF(F(F(F(" }{TEXT -1 2 "= " }{XPPEDIT 18 0 "matrix([[0, 0, 2^(2*k), 2^(2*k)], [0, 0, 2^(2*k), 2^(2*k)], [2^( 2*k), 2^(2*k), 0, 0], [2^(2*k), 2^(2*k), 0, 0]]);" "6#-%'matrixG6#7&7& \"\"!F()\"\"#*&F*\"\"\"%\"kGF,)F**&F*F,F-F,7&F(F()F**&F*F,F-F,)F**&F*F ,F-F,7&)F**&F*F,F-F,)F**&F*F,F-F,F(F(7&)F**&F*F,F-F,)F**&F*F,F-F,F(F( " }{TEXT 260 3 "(1)" }{TEXT -1 23 " \n\nalors montrons que " } {XPPEDIT 18 0 "A^(2*k+3);" "6#)%\"AG,&*&\"\"#\"\"\"%\"kGF(F(\"\"$F(" } {TEXT -1 1 "=" }{XPPEDIT 18 0 "matrix([[0, 0, 2^(2*k+2), 2^(2*k+2)], [ 0, 0, 2^(2*k+2), 2^(2*k+2)], [2^(2*k+2), 2^(2*k+2), 0, 0], [2^(2*k+2), 2^(2*k+2), 0, 0]]);" "6#-%'matrixG6#7&7&\"\"!F()\"\"#,&*&F*\"\"\"%\"k GF-F-F*F-)F*,&*&F*F-F.F-F-F*F-7&F(F()F*,&*&F*F-F.F-F-F*F-)F*,&*&F*F-F. F-F-F*F-7&)F*,&*&F*F-F.F-F-F*F-)F*,&*&F*F-F.F-F-F*F-F(F(7&)F*,&*&F*F-F .F-F-F*F-)F*,&*&F*F-F.F-F-F*F-F(F(" }{TEXT -1 3 "(2)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "D\351monstration :" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{XPPEDIT 18 0 "A^(2*k+3);" "6#)%\"AG,&*&\"\"#\" \"\"%\"kGF(F(\"\"$F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A^(2*k+1);" "6 #)%\"AG,&*&\"\"#\"\"\"%\"kGF(F(F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "A ^2;" "6#*$%\"AG\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[0, 0, 2^(2*k), 2^(2*k)], [0, 0, 2^(2*k), 2^(2*k)], [2^(2*k), 2^(2*k), 0, 0] , [2^(2*k), 2^(2*k), 0, 0]]);" "6#-%'matrixG6#7&7&\"\"!F()\"\"#*&F*\" \"\"%\"kGF,)F**&F*F,F-F,7&F(F()F**&F*F,F-F,)F**&F*F,F-F,7&)F**&F*F,F-F ,)F**&F*F,F-F,F(F(7&)F**&F*F,F-F,)F**&F*F,F-F,F(F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "A^2;" "6#*$%\"AG\"\"#" }{TEXT -1 1 " " }{TEXT 262 37 ", par l'hypoth\350se (1) de l'induction." }{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 40 "Calculons le membre de droite avec Mapl e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "C:=matrix(4,4,[0,0,2^( 2*k),2^(2*k),0,0,2^(2*k),2^(2*k),2^(2*k),2^(2*k),0,0,2^(2*k),2^(2*k),0 ,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Multiplions C par A^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A^(2*k+3)=simplify(multip ly(C,A^2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "On a bien le r\351 sultat (2) car " }{XPPEDIT 18 0 "4^(1+k);" "6#)\"\"%,&\"\"\"F&%\"kGF& " }{TEXT -1 2 "= " }{XPPEDIT 18 0 "(2^2)^(1+k);" "6#)*$\"\"#F%,&\"\"\" F'%\"kGF'" }{TEXT -1 3 "= " }{XPPEDIT 18 0 "2^(2*k+2)" "6#)\"\"#,&*&F $\"\"\"%\"kGF'F'F$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Dans l'exercice 3) vous aurez \+ \340 compl\351ter la preuve en faisant, \340 l'aide de Maple, le cas B ) : " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 12 "Exercices : " }{TEXT -1 0 "" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 284 5 "No 1)" }{TEXT 286 1 " " }{TEXT 273 39 "Pour chacune des matrices A suivantes, " }}{PARA 256 "" 0 "" {TEXT 291 2 "A)" }{TEXT 293 57 " Trouver une fo rmule pour A^n o\371 n est un nombre naturel " }}{PARA 256 "" 0 "" {TEXT 292 3 "B) " }{TEXT 294 26 "Prouvez-la par r\351currence " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 285 2 "A)" }{TEXT -1 10 " Soit A = " } {XPPEDIT 18 0 "matrix([[a, 0], [0, b]])" "6#-%'matrixG6#7$7$%\"aG\"\"! 7$F)%\"bG" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 295 32 " Es pace de travail de l'\351tudiant" }{TEXT -1 0 "" }}{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 274 2 "B)" }{TEXT -1 10 " Soit A = " }{XPPEDIT 18 0 "matrix ([[1, 0], [1, 1]]);" "6#-%'matrixG6#7$7$\"\"\"\"\"!7$F(F(" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 296 32 " Espace de travail de l'\351tudiant" } {TEXT -1 0 "" }}{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 282 2 "C)" }{TEXT -1 10 " soit A = " }{XPPEDIT 18 0 "matrix([[a, b], [0, 0]]);" "6#-%'ma trixG6#7$7$%\"aG%\"bG7$\"\"!F+" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 297 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 2 "D)" }{TEXT 275 1 " " }{TEXT -1 9 "Soit A = " }{XPPEDIT 18 0 "matrix([[1, 0, 0], [1, -1, 0], [1, 1, 1] ])" "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F(,$F(!\"\"F)7%F(F(F(" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 298 32 " Espace de travail de l'\351tudiant" } {TEXT -1 0 "" }}{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 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 1 " \+ " }{TEXT 276 2 "E)" }{TEXT 277 1 " " }{TEXT -1 10 " Soit A = " } {XPPEDIT 18 0 "matrix([[2, 0, 0, 2], [0, 3, 3, 0], [0, 3, 3, 0], [2, 0 , 0, 2]]);" "6#-%'matrixG6#7&7&\"\"#\"\"!F)F(7&F)\"\"$F+F)7&F)F+F+F)7& F(F)F)F(" }{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 299 32 " Esp ace de travail de l'\351tudiant" }{TEXT -1 0 "" }}{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 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 1 " " }{TEXT 278 2 "F)" }{TEXT 279 1 " " }{TEXT -1 10 " Soit A = " }{XPPEDIT 18 0 "matrix([[0, 1, 0, 1], [1, 0, 1, 0], [0, 1, 0, 1], [1, 0, 1, 0]]);" "6#-%'matrixG6#7&7&\"\"!\"\"\"F(F)7&F) F(F)F(7&F(F)F(F)7&F)F(F)F(" }{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 300 32 " Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }} {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 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 2 "G)" }{TEXT 280 1 " " }{TEXT -1 10 " Soit A = " }{XPPEDIT 18 0 "matrix([[0, 1, 0], [1, 0, 3], [0, 1, 0]]);" "6#-% 'matrixG6#7%7%\"\"!\"\"\"F(7%F)F(\"\"$7%F(F)F(" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 301 32 " Espace de travail de l'\351tudiant" }{TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 289 5 "No 2)" }{TEXT 290 1 " " }{TEXT 288 84 "P our la matrice A suivante, trouver une formule pour A^n o\371 n est un nombre naturel " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 2 "H)" }{TEXT 281 2 "* " }{TEXT -1 9 "Soit A = " } {XPPEDIT 18 0 "matrix([[1, 0, -1], [0, 1/2, 0], [1, 0, 1]]);" "6#-%'ma trixG6#7%7%\"\"\"\"\"!,$F(!\"\"7%F)*&F(F(\"\"#F+F)7%F(F)F(" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 302 32 " Espace de travail de l'\351tudiant" } {TEXT -1 0 "" }}{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 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT 283 5 "No 3)" }{TEXT 287 1 " " }{TEXT -1 95 "Com pl\350ter la preuve par r\351currence de l'exemple 2), en faisant, \+ \340 l'aide de Maple, le cas B) : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "Si l'exposant de A est pair, prouvez que \+ pour tout n, " }{XPPEDIT 18 0 "A^(2*n) = matrix([[2^(2*n-1), 2^(2*n-1 ), 0, 0], [2^(2*n-1), 2^(2*n-1), 0, 0], [0, 0, 2^(2*n-1), 2^(2*n-1)], \+ [0, 0, 2^(2*n-1), 2^(2*n-1)]]);" "6#/)%\"AG*&\"\"#\"\"\"%\"nGF(-%'matr ixG6#7&7&)F',&*&F'F(F)F(F(F(!\"\")F',&*&F'F(F)F(F(F(F2\"\"!F67&)F',&*& F'F(F)F(F(F(F2)F',&*&F'F(F)F(F(F(F2F6F67&F6F6)F',&*&F'F(F)F(F(F(F2)F', &*&F'F(F)F(F(F(F27&F6F6)F',&*&F'F(F)F(F(F(F2)F',&*&F'F(F)F(F(F(F2" }} {SECT 1 {PARA 4 "" 0 "" {TEXT 303 32 " Espace de travail de l'\351tudi ant" }{TEXT -1 0 "" }}{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 "" {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 }