{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 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 256 "" 0 "" {TEXT 256 34 "Syst\350mes lin\351aire s et d\351terminants" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wit h(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected name s norm and trace have been redefined and unprotected\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "mat1:=matrix(5,5,[[-2,1,3,1,-2],[-2,-2,1,3,1],[1,-2,- 2,1,3],[3,1,-2,-2,1],[1,3,1,-2,-2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mat1G-%'matrixG6#7'7'!\"#\"\"\"\"\"$F+F*7'F*F*F+F,F+7'F+F*F*F+F, 7'F,F+F*F*F+7'F+F,F+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "inverse(mat1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\" \"\"F(!\"#\"\"$F)7'F)F(F(F)F*7'F*F)F(F(F)7'F)F*F)F(F(7'F(F)F*F)F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "det(mat1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "C'est ce qui explique le r\351sultat si simple (sans fraction)..." }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 2" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "matA:=matrix(3,3,[[1,1,m],[1,1,-1],[1,m,-1]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%matAG-%'matrixG6#7%7%\"\"\"F*% \"mG7%F*F*!\"\"7%F*F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " factor(det(matA));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&!\"\"\"\"\"% \"mGF&F&,&F'F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sol ve(\{x+y+m*z=m,x+y-z=1,x+m*y-z=1\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG,$*&%\"mG\"\"\",&F(F)F)F)!\"\"\"\"#/%\"zG*&,&F+ F)F(F)F)F*F+/%\"yG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "En fai t, c'est lorsque le syst\350me est de Cramer; il reste faire les cas \+ particuliers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "m:=1:solve (\{x+y+m*z=m,x+y-z=1,x+m*y-z=1\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yGF%/%\"zG\"\"!/%\"xG,&\"\"\"F,F%!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "m:=-1:solve(\{x+y+m*z=m,x+y- z=1,x+m*y-z=1\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "C ela signifie non pas que Maple ne trouve pas de solution, mais qu'il n 'en existe pas. Maple sait en effet r\351soudre les syst\350mes lin \351aires..." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 3" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?colspace" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "The functions rowspace(A) and colspace(A) return a set of vectors that form a basis for the vector space spanned by th e rows and columns of the matrix A, respectively. The vectors are retu rned in a canonical form with leading entries 1. " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 68 "mat31:=matrix(4,1,[[0],[1],[1],[0]]);\nkerne l(mat31),colspace(mat31);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat31G -%'MATRIXG6#7&7#\"\"!7#\"\"\"F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$< \"<#-%'VECTORG6#7&\"\"!\"\"\"F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "mat32:=matrix(1,4,[[0,1,1,0]]);\nkernel(mat32),colspa ce(mat32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat32G-%'MATRIXG6#7#7 &\"\"!\"\"\"F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<%-%'VECTORG6#7&\" \"\"\"\"!F)F)-F%6#7&F)!\"\"F(F)-F%6#7&F)F)F)F(<#-F%6#7#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "mat33:=matrix(3,4,[[0,1,2,0],[1,0,1 ,0],[0,1,2,0]]);\nkernel(mat33);colspace(mat33);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat33G-%'MATRIXG6#7%7&\"\"!\"\"\"\"\"#F*7&F+F*F+F*F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7&\"\"!F(F(\"\"\"-F% 6#7&F)\"\"#!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7% \"\"!\"\"\"F(-F%6#7%F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "mat34:=matrix(3,2,[[1,2],[3,4],[5,6]]);\nkernel(mat34);colspace(ma t34);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat34G-%'MATRIXG6#7%7$\"\" \"\"\"#7$\"\"$\"\"%7$\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7%\"\"!\"\"\"\"\"#-F %6#7%F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "mat35:=ma trix(2,3,[[1,2,3],[4,5,6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat 35G-%'MATRIXG6#7$7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "kernel(mat35);colspace(mat35);rowspace(ma t35);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'VECTORG6#7%\"\"\"!\"#F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7$\"\"!\"\"\"-F%6#7$ F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7%\"\"\"\"\"!!\" \"-F%6#7%F)F(\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice \+ 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "mat41:=matrix(2,2,[[cos (theta),-sin(theta)],[sin(theta),cos(theta)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat41G-%'matrixG6#7$7$-%$cosG6#%&thetaG,$-%$sinGF,! \"\"7$F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "kernel(mat41) ,colspace(mat41);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<\"<$-%'vectorG6# 7$\"\"\"\"\"!-F&6#7$F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "det(mat41);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)-%$cosG6#%&thetaG \"\"#\"\"\"F+*$)-%$sinGF(F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "mat42:=matrix(3,3,[[1,1,0],[ 2,2,1],[1,0,-2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat42G-%'matr ixG6#7%7%\"\"\"F*\"\"!7%\"\"#F-F*7%F*F+!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "kernel(mat42),colspace(mat42);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$<\"<%-%'vectorG6#7%\"\"!\"\"\"F)-F&6#7%F)F)F*-F&6#7%F *F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "mat43:=matrix(3,3, [[1,1,0],[2,2,0],[1,0,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat 43G-%'matrixG6#7%7%\"\"\"F*\"\"!7%\"\"#F-F+7%F*F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "kernel(mat43),colspace(mat43);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$<#-%'vectorG6#7%\"\"\"!\"\"F(<$-F%6#7% F(\"\"#\"\"!-F%6#7%F/F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "mat44:=matrix(4,4,[[1,1,2,0],[0,2,-1,1],[0,0,3,2],[0,0,0,-1]]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat44G-%'matrixG6#7&7&\"\"\"F*\"\" #\"\"!7&F,F+!\"\"F*7&F,F,\"\"$F+7&F,F,F,F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "kernel(mat44),colspace(mat44);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<\"<&-%'vectorG6#7&\"\"!F)\"\"\"F)-F&6#7&F)F)F)F*-F&6#7 &F*F)F)F)-F&6#7&F)F*F)F)" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exe rcice 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "mat_5 := matrix( [[1, cos(t_1), cos(2*t_1)], [1, cos(t_2), cos(2*t_2)], [1, cos(t_3), c os(2*t_3)]]);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat_5G-%'matri xG6#7%7%\"\"\"-%$cosG6#%$t_1G-F,6#,$F.\"\"#7%F*-F,6#%$t_2G-F,6#,$F6F27 %F*-F,6#%$t_3G-F,6#,$F=F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "factor(det(mat_5));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&-%$cosG 6#%$t_2G\"\"\"-F&6#,$%$t_3G\"\"#F)F)*&-F&6#,$F(F.F)-F&6#F-F)!\"\"*&-F& 6#%$t_1GF)F*F)F5*&-F&6#,$F9F.F)F3F)F)*&F7F)F0F)F)*&F;F)F%F)F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Gloups ..." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "combine(%,trig);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:-%$cosG6#,&%$t_2G!\"\"*&\"\"#\"\"\"%$t_3GF,F,#F,F+*&F.F,-F%6#, &F(F,*&F+F,F-F,F,F,F,*&#F,F+F,-F%6#,&F(!\"#F-F,F,F)*&#F,F+F,-F%6#,&F(F +F-F,F,F)*&#F,F+F,-F%6#,&%$t_1GF)*&F+F,F-F,F,F,F)*&#F,F+F,-F%6#,&FDF,* &F+F,F-F,F,F,F)*&F.F,-F%6#,&FDF9F-F,F,F,*&F.F,-F%6#,&FDF+F-F,F,F,*&F.F ,-F%6#,&FDF)*&F+F,F(F,F,F,F,*&F.F,-F%6#,&FDF,*&F+F,F(F,F,F,F,*&#F,F+F, -F%6#,&FDF9F(F,F,F)*&#F,F+F,-F%6#,&FDF+F(F,F,F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# ,.*&-%$cosG6#%$t_2G\"\"\")-F&6#%$t_3G\"\"#F)F.*(F.F)F+F))F%F.F)!\"\"*( F.F)-F&6#%$t_1GF)F*F)F1*(F.F)F+F))F3F.F)F)*(F.F)F%F)F7F)F1*(F.F)F3F)F0 F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&-%$cosG6#%$t_3G\"\"\"-F'6#%$t_2G !\"\"F*,&-F'6#%$t_1GF*F+F.F*,&F0F*F&F.F*\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Sans les mains !" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "matA:=m atrix(3,3,[[2*(a-1),2,-1],[2,2*a,2],[4*a,2*(2*a+1),2*a+1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%matAG-%'matrixG6#7%7%,&%\"aG\"\"#F,!\"\"F ,F-7%F,,$F+F,F,7%,$F+\"\"%,&F+F2F,\"\"\",&F+F,F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(det(matA));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"aG\"\"\",&F%\"\"#F&!\"\"F&,&F%F&F&F)F&\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "solve(\{2*(a-1)*x+2*y-z=2*( a+1),2*x+2*a*y+2*z=4*a^2+3,4*a*x+2*(2*a+1)*y+(2*a+1)*z=16*a^3-2*a^2-a+ 5\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG,$*&,(%\"aG \"#M\"#>\"\"\"*&\"#;F,)F)\"\"#F,F,F,,&F)F0F,!\"\"F2#F,F0/%\"yG,&F)!\"' \"\"$F2/%\"zG*&,**$F/F,!#5*&\"# " 0 "" {MPLTEXT 1 0 15 "solve(a*x =a,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 50 "Pour a=0, tous les r\351els sont pourtant solution ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "a:=0:solve(\{2*(a-1 )*x+2*y-z=2*(a+1),2*x+2*a*y+2*z=4*a^2+3,4*a*x+2*(2*a+1)*y+(2*a+1)*z=16 *a^3-2*a^2-a+5\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\" zGF%/%\"yG,&#\"\"&\"\"#\"\"\"*&#F,F+F,F%F,!\"\"/%\"xG,&#\"\"$F+F,F%F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "a:=1:solve(\{2*(a-1)*x +2*y-z=2*(a+1),2*x+2*a*y+2*z=4*a^2+3,4*a*x+2*(2*a+1)*y+(2*a+1)*z=16*a^ 3-2*a^2-a+5\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"zGF %/%\"xG,&#\"\"$\"\"#\"\"\"*&#F*F+F,F%F,!\"\"/%\"yG,&F+F,*&#F,F+F,F%F,F ," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "a:=1/2:solve(\{2*(a-1 )*x+2*y-z=2*(a+1),2*x+2*a*y+2*z=4*a^2+3,4*a*x+2*(2*a+1)*y+(2*a+1)*z=16 *a^3-2*a^2-a+5\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "P as de solution..." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 8 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "?vandermonde" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "The function vandermonde(L) returns the \+ Vandermonde matrix formed from the elements of the list. This square m atrix has as its (i,j)-th entry L[i]^(j-1). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "VDM:=vandermonde([1515,exp(-1),pi]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$VDMG-%'matrixG6#7%7%\"\"\"\"%::\"(D_H#7%F *-%$expG6#!\"\"*$)F.\"\"#F*7%F*%#piG*$)F6F4F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "VDM2:=vandermonde([lambda1,lambda2,lambda3,lambd a4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VDM2G-%'matrixG6#7&7&\"\" \"%(lambda1G*$)F+\"\"#F**$)F+\"\"$F*7&F*%(lambda2G*$)F3F.F**$)F3F1F*7& F*%(lambda3G*$)F9F.F**$)F9F1F*7&F*%(lambda4G*$)F?F.F**$)F?F1F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "factor(det(VDM));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&!%::\"\"\"-%$expG6#!\"\"F&F&,&F%F&%#piGF &F&,&F,F&F'F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(d et(VDM2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,&%(lambda1G!\"\"%(lam bda4G\"\"\"F(,&F%F&%(lambda2GF(F(,&F*F(F'F&F(,&F%F&%(lambda3GF(F(,&F-F (F'F&F(,&F-F(F*F&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Une petite id\351e de la formule g\351n\351rale ?" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 9" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f un_9 :=(i,j) -> if j=i then (1+x) else 1 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&fun_9Gf*6$%\"iG%\"jG6\"6$%)operatorG%&arrowGF)@%/9%9 $,&\"\"\"F2%\"xGF2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mat_9 := n -> matrix(n,n,fun_9);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&mat_9Gf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%'matrixG6%9$F/%&fun_9G F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "mat_9(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&,&\"\"\"F)%\"xGF)F)F)F )7&F)F(F)F)7&F)F)F(F)7&F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq(factor(det(mat_11(i))),i=2..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'*&%\"xG\"\"\",&\"\"#F%F$F%F%*&)F$F'F%,&\"\"$F%F$F%F%*&) F$F+F%,&\"\"%F%F$F%F%*&)F$F/F%,&\"\"&F%F$F%F%*&)F$F3F%,&\"\"'F%F$F%F% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "La encore, une g\351niale int uition peut nous donner une petite id\351e du r\351sultat..." }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 10" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 106 "fun_10:=(i,j) -> if j=i then if i=1 then cos( theta) else 2*cos(theta) fi elif abs(j-i)=1 then 1 else 0 fi;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%'fun_10Gf*6$%\"iG%\"jG6\"6$%)operato rG%&arrowGF)@'/9%9$@%/F0\"\"\"-%$cosG6#%&thetaG,$F4\"\"#/-%$absG6#,&F/ F3F0!\"\"F3F3\"\"!F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "mat_10 := n-> matrix(n,n,fun_10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'mat_10Gf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%'matrixG6%9$F/%'fun_10G F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mat_10(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&-%$cosG6#%&thetaG\"\" \"\"\"!F-7&F,,$F(\"\"#F,F-7&F-F,F/F,7&F-F-F,F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "seq(combine(det(mat_10(i)),trig),i=2..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'-%$cosG6#,$%&thetaG\"\"#-F$6#,$F'\"\"$ -F$6#,$F'\"\"%-F$6#,$F'\"\"&-F$6#,$F'\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Hum..." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice \+ 11" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:='a':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "fun_11 := (i,j) -> if j=i then a+b \+ else if j=i+1 then b else if j=i-1 then a else 0 fi fi fi;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'fun_11Gf*6$%\"iG%\"jG6\"6$%)operatorG%&ar rowGF)@%/9%9$,&%\"aG\"\"\"%\"bGF3@%/F/,&F0F3F3F3F4@%/F/,&F0F3F3!\"\"F2 \"\"!F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mat_11:= n-> matrix(n,n,fun_11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'mat_11Gf*6# %\"nG6\"6$%)operatorG%&arrowGF(-%'matrixG6%9$F/%'fun_11GF(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mat_11(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&,&%\"aG\"\"\"%\"bGF*F+\"\"!F,7&F)F(F +F,7&F,F)F(F+7&F,F,F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " seq(det(mat_11(i)),i=2..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,(*$)% \"aG\"\"#\"\"\"F(*&%\"bGF(F&F(F(*$)F*F'F(F(,**$)F&\"\"$F(F(*&F*F(F%F(F (*&F&F(F,F(F(*$)F*F0F(F(,,*$)F&\"\"%F(F(*&F*F(F/F(F(*&F%F(F,F(F(*&F&F( F4F(F(*$)F*F8F(F(,.*$)F&\"\"&F(F(*&F*F(F7F(F(*&F/F(F,F(F(*&F%F(F4F(F(* &F&F(F=F(F(*$)F*FAF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "d et(mat_11(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$)%\"aG\"#5\"\" \"F(*&)F&\"\"*F(%\"bGF(F(*&)F&\"\")F()F,\"\"#F(F(*&)F&\"\"(F()F,\"\"$F (F(*&)F&\"\"'F()F,\"\"%F(F(*&)F&\"\"&F()F,F>F(F(*&)F,F9F()F&F;F(F(*&)F ,F4F()F&F6F(F(*&)F,F/F()F&F1F(F(*&)F,F+F(F&F(F(*$)F,F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify((a^11-b^11)/(a-b));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$)%\"aG\"#5\"\"\"F(*&)F&\"\"*F(%\"b GF(F(*&)F&\"\")F()F,\"\"#F(F(*&)F&\"\"(F()F,\"\"$F(F(*&)F&\"\"'F()F,\" \"%F(F(*&)F&\"\"&F()F,F>F(F(*&)F,F9F()F&F;F(F(*&)F,F4F()F&F6F(F(*&)F,F /F()F&F1F(F(*&)F,F+F(F&F(F(*$)F,F'F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 12" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " mat12:=matrix(4,4,[[a,b,c,d],[b,a,d,c],[c,d,a,b],[d,c,b,a]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mat12G-%'matrixG6#7&7&%\"aG%\"bG%\" cG%\"dG7&F+F*F-F,7&F,F-F*F+7&F-F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "factor(det(mat12));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#**,*%\"bG\"\"\"%\"aGF&%\"cGF&%\"dGF&F&,*F%F&F'F&F(!\"\"F)F+F&,*F%F+F 'F&F(F&F)F+F&,*F%F+F'F&F(F+F)F&F&" }}}}}{MARK "12 2 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }