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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 259 "" 0 "" {TEXT 341 25 "Laboratoire 4 - Matric es" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 52 "Par Claude St-Hilaire, cl aude.sthilaire@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 343 51 "Pr incipales commandes utilis\351es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "array, matrix de la biblioth\350que linalg, ra ndmatrix, piecewise, row,col, delrows, delcols, submatrix, augment, ev alm, multiply, dotprod, subs avec op, equate avec la biblioth\350que s tudent, solve, seq, diag, identity, symmetric, antisymmetric, rand." } }{PARA 0 "" 0 "" {TEXT -1 69 "Note : S\351lectionner un mot et utilise r l'aide pour plus d'information" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 45 "D\351finitions de matrice : avec matrix ou array" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Les matrices sont un cas particulier de l a commande Maple " }{TEXT 259 7 "array :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "array(1..2,1..4,[[2,-12,45,-122],[3,40,-245,99]]);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Cependant les matrices peuvent \+ \352tre d\351finies par la commande Maple " }{TEXT 257 6 "matrix" } {TEXT -1 20 " de la biblioth\350que " }{TEXT 268 6 "linalg" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Premi\350re mani\350re ave c matrix:" }{TEXT 258 1 " " }{TEXT 363 14 "matrix(m,n,[ a" }{TEXT 260 3 "1,1" }{TEXT 261 3 ", a" }{TEXT 262 3 "1,2" }{TEXT 263 6 " ... a" } {TEXT 264 3 "m,n" }{TEXT 265 3 "]);" }{TEXT -1 57 " ou m est le nombre de lignes et n, le nombre de colonnes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "matrix(2,3,[3,6,40,-9,4,7]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Deuxi\350me mani\350re : " }{TEXT 266 10 "matrix([[a" }{TEXT 269 3 "1,1" }{TEXT 270 3 ", a" }{TEXT 271 4 "1,2 " }{TEXT 272 5 "... a" }{TEXT 273 3 "1,n" }{TEXT 274 5 "], [a" }{TEXT 275 3 "2,1" } {TEXT 276 3 ", a" }{TEXT 277 3 "2,2" }{TEXT 278 6 " ... a" }{TEXT 279 3 "2,n" }{TEXT 280 7 "]... [a" }{TEXT 281 3 "m,1" }{TEXT 282 3 ", a" } {TEXT 283 3 "m,2" }{TEXT 284 6 " ... a" }{TEXT 285 3 "m,n" }{TEXT 286 4 "]]);" }{TEXT -1 44 "\ncr\351e une matrice \340 m lignes et n colonn es. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "matrix([[1,2,3,4],[ 12,23,45,67],[-1,7,23,100]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 " Troisi\350me mani\350re : " }{TEXT 267 14 "matrix(m,n,[[a" }{TEXT 287 3 "1,1" }{TEXT 288 3 ", a" }{TEXT 289 4 "1,2 " }{TEXT 290 5 "... a" } {TEXT 291 3 "1,n" }{TEXT 292 5 "], [a" }{TEXT 293 3 "2,1" }{TEXT 294 3 ", a" }{TEXT 295 3 "2,2" }{TEXT 296 6 " ... a" }{TEXT 297 3 "2,n" } {TEXT 298 7 "]... [a" }{TEXT 299 3 "m,1" }{TEXT 300 3 ", a" }{TEXT 301 3 "m,2" }{TEXT 302 6 " ... a" }{TEXT 303 3 "m,n" }{TEXT 304 4 "]]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "matrix(3,2,[[22,55],[- 1,0],[333,-876]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "On peut aus si d\351finir une matrice avec des entr\351es " }{TEXT 306 10 "al\351a toires" }{TEXT -1 26 " entiers de -99 \340 99 avec " }{TEXT 305 16 "ra ndmatrix(m,n) " }{TEXT 323 26 "qui cr\351e une matrice (mxn)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "randmatrix(5,4);# donne une \+ matrice 5x4 avec des entr\351es al\351atoires" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "randmatrix(m,n,entries = rand(a..b)); retourne des ent iers entre a et b inclus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "randmatrix(2,3,entries = rand(1..10));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "On peut aussi d\351finir une matrice A \340 l'aide d'une \+ fonction : Exemple : si on veut ai,j = i^2 + j^2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=matrix(4,4,(i,j)->i^2+j^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "matrix(3,4,(i,j)->piecewise(i<=j,i+ j,0));" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 307 53 "Les \351l\351ments d es matrices, les lignes et les colonnes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 308 47 "A[i,j] d\351signe l'\351l\351ment ai,j d e la matrice A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A[3,4];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Si on encadre une expression ma th\351matique de guillemets, Maple l'affiche sans la calculer" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "'A[2,2]'=A[2,2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 310 9 "row(M,i) " }{TEXT -1 39 "extraie la i\350 me ligne de la matrice M \n" }{TEXT 311 8 "col(M,j)" }{TEXT -1 41 " ex traie la j\350me colonne de la matrice M." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "row(A,1);col(A,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "B:=randmatrix(4,5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 11 "row(M,i..j)" }{TEXT -1 46 " extraie les lignes de i \340 j de \+ la matrice M \n" }{TEXT 313 11 "col(M,i..j)" }{TEXT -1 48 ", extraie l es colonnes de i \340 j de la matrice M." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "col(B,1..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "row(B,2..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 344 7 "delrows" } {TEXT -1 4 " et " }{TEXT 345 7 "delcols" }{TEXT -1 73 " cr\351e une ma trice o\371 on a effac\351 des lignes ou des colonnes d'une matrice" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "delrows(B,2..3);# matrice \+ obtenue en enlevant les lignes 2 \340 3 de la matrice B." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "delcols(B,2..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "On peut extraire une sous-matrice" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "submatrix(B,2..3,2..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "***On peut ajouter des colonnes \340 une \+ matrice avec la commande " }{TEXT 309 7 "augment" }{TEXT -1 78 " (ou c oncat) : ajoutons les 2 premi\350res colonnes de la matrice A la matr ice B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "C:=augment(B,col(A ,1..2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "On peut ajouter des l ignes (ou une matrice) sous une matrice avec la commande stackmatrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "stackmatrix(C,vector([1,2 ,3,4,5,6,7]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 314 20 "Alg\350bre des matrices" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A:=matrix([[2,7,9],[-1,10,15 ],[4,-1,0]]);B:=matrix([[5,16,7],[4,-5,8],[10,20,30]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 315 35 "Addition et soustraction de matrice" } {TEXT -1 1 "s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A+B;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "C:=evalm(A+B);#evalm pour af ficher les valeurs de A+B.(ou matadd(A,B)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 24 "C[i,j] = A[i,j] + B[i,j]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A[2,3];B[2,3];C[2,3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "On a : C[2,3] = A[2,3] + B[2,3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "C=evalm(A-B);#C[i,j] = A[i,j] - B[i,j]" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 61 "Multiplication d'une matrice M p ar un scalaire k :evalm( k*M)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C:=evalm(10*B);# ou scalarmul(B,10);# C = k*B,k r\351el : " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 356 16 "C[i,j]= k*B[i,j]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Pour une combinaison lin\351aire de matr ices xA + yB - zC : evalm(x*A + y*B + z*C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalm(3*A+4*B-5*C);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 67 "Multiplication de 2 matrices A et B : multiply(A,B) ou ev alm(A&*B) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "C:=multiply(A ,B):'A'=evalm(A),'B'=evalm(B),'C'=evalm(C), 'C=AB';" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Soit A(mxn), B(nxp) et C(mxp) = AB; " }{TEXT 330 8 "C[i,j] =" }{XPPEDIT 18 0 "Sum(A[i,k]*B[k,j],k = 1 .. n);" "6#-% $SumG6$*&&%\"AG6$%\"iG%\"kG\"\"\"&%\"BG6$F+%\"jGF,/F+;F,%\"nG" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 346 11 " Exemple 1 :" }{TEXT -1 10 " C[2,3] = " }{XPPEDIT 18 0 "Sum(A[2,k]*B[k, 3],k = 1 .. 3);" "6#-%$SumG6$*&&%\"AG6$\"\"#%\"kG\"\"\"&%\"BG6$F+\"\"$ F,/F+;F,F0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A[2,1]*B[1,3]+A[2,2]*B[2 ,3]+A[2,3]*B[3,3];" "6#,(*&&%\"AG6$\"\"#\"\"\"F)&%\"BG6$F)\"\"$F)F)*&& F&6$F(F(F)&F+6$F(F-F)F)*&&F&6$F(F-F)&F+6$F-F-F)F)" }{TEXT -1 33 " = (- 1)(7)+(10)(8)+(15)(30) = 523" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "C[2,3]=(-1)*(7)+(10)*(8)+(15)*(30);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "En consid\351rant la (i\350me ligne de A) et la (j i\350 me colonne de B) comme des vecteurs-lignes, alors C[i,j] se calcule co mme le produit scalaire de : (i\350me ligne de A) et (j i\350me colonn e de B)" }}{PARA 0 "" 0 "" {TEXT 331 48 "C[i,j] = (i\350me ligne de A) o(j i\350me colonne de B)" }{TEXT -1 16 " ou en Maple : " }{TEXT 332 26 "dotprod(row(A,i),col(B,j))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "row(A,2),col(B,3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "'C[2,3]'=dotprod(row(A,2),co l(B,3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(li nalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "A:=matrix(2,2,[a ,2,3,b]):B:=matrix(2,2,[3,1,2,10]):C:=matrix([[-146, -30], [45, 183]]) :'A'=evalm(A),'B'=evalm(B),'C'=evalm(C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 359 11 "Exemple 2 :" }{TEXT -1 89 " Trouver a et b en utilisant \+ des produits scalaires de lignes avec des colonnes si C = AB" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "C[2,1]=dotprod(row(A,2),col( B,1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b=solve(%,b);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "C[1,2]=dotprod(row(A,1),col (B,2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a=solve(%,a);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Autre solution : Avec " }{TEXT 360 6 "equate" }{TEXT -1 20 " de la biblioth\350que " }{TEXT 361 7 "st udent" }{TEXT -1 23 " et solve. La commande " }{TEXT 362 11 "equate(A, B)" }{TEXT -1 92 " \351galise terme \340 terme, les matrices A et B. N ote : equate sert aussi \340 \351galiser 2 vecteurs " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(student,equate):equate(multiply(A,B) ,C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(%,\{a,b\});" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 324 29 "Substitution dans une matric e" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A:=matri x(2,2,[3*x,y+x,x^2,x*y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=5,y=10\},A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Nous devons indiquer \340 Maple de substituer sur les composantes (" } {TEXT 325 2 "op" }{TEXT -1 23 "\351rands) de la matrice A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(\{x=5,y=10\},op(A));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT 316 20 "\311galit\351 de matrices " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "R\351soudre A = B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A:=matrix(3,2,[a,2,2*b-1,c,-7,c*d]);B:=matrix(3,2,[2*a,2,c,3,- 7,b+c]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Premi\350re mani\350r e : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(student,equate ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "equate(A,B);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "On r\351soud avec solve" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(%,\{a,b,c,d\});" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Deuxi\350me mani\350re : Pour r \351soudre A=B, on doit r\351soudre A[i,j]=B[i,j] pour i=1..3,j=1..2. \+ Cela peut se faire avec la commande seq suivie de la commande solve : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "X:=seq(seq(A[i,j]=B[i,j ],i=1..3),j=1..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve (\{X\},\{a,b,c,d\});" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 342 56 "Combin aison lin\351aire des vecteurs-colonnes d'une matrice" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 154 "Soit A une matrice dont les colonnes sont a1 , a2, a3 i.e. A = [a1, a2,a3] et x, la matrice-colonne x = (x1,x2,x3) . V\351rifions que Ax = x1a1 + x2a2 + x3a3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "a1:=matrix(3,1,[1,2,3]):a2:=matrix(3,1,[5,7,9]):a 3:=matrix(3,1,[10,20,30]):a1=evalm(a1),a2=evalm(a2),a3=evalm(a3);A:=au gment(a1,a2,a3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x:=matr ix(3,1,[x1,x2,x3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mult iply(A,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(x1*a1+x 2*a2+x3*a3);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "matrix([[1, 5, \+ 10], [2, 7, 20], [3, 9, 30]]);" "6#-%'matrixG6#7%7%\"\"\"\"\"&\"#57%\" \"#\"\"(\"#?7%\"\"$\"\"*\"#I" }{XPPEDIT 18 0 "matrix([[x1], [x2], [x3] ])" "6#-%'matrixG6#7%7#%#x1G7#%#x2G7#%#x3G" }{TEXT -1 5 " = x1" } {XPPEDIT 18 0 "matrix([[1], [2], [3]]);" "6#-%'matrixG6#7%7#\"\"\"7#\" \"#7#\"\"$" }{TEXT -1 5 " + x2" }{XPPEDIT 18 0 "matrix([[5], [7], [9]] );" "6#-%'matrixG6#7%7#\"\"&7#\"\"(7#\"\"*" }{TEXT -1 5 " + x3" } {XPPEDIT 18 0 "matrix([[10], [20], [30]]);" "6#-%'matrixG6#7%7#\"#57# \"#?7#\"#I" }{TEXT -1 3 " , " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "D onc si a1, a2.. an sont les colonnes de A(mxn), A = [a1,a2,..,an] et x = " }{XPPEDIT 18 0 "matrix([[x1], [x2], [_], [xn]]);" "6#-%'matrixG6# 7&7#%#x1G7#%#x2G7#%\"_G7#%#xnG" }{TEXT -1 34 " alors Ax = x1a1 + x2a2 \+ + ..+ xnan" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 326 30 "Matrices carr\351es particuli\350res" }} {EXCHG {PARA 0 "" 0 "" {TEXT 334 17 "Matrice diagonale" }{TEXT -1 20 " : diag(a1,a2,..,an)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dia g(1,-20,111);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "La " }{TEXT 335 20 "matrice nulle 0(nxn)" }{TEXT -1 3 " : " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "array(sparse,1..3,1..3);# ou matrix(3,3,(i,j)->0); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "La " }{TEXT 336 24 "matrice id entit\351 I(nxn) " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "I = array(identity,1..3,1..3);# ou diag(1,1,1) ou mat rix(3,3,(i,j)->piecewise(i=j,1,0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Pour une " }{TEXT 337 22 "matrice antisym\351trique" }{TEXT -1 56 " avec des nombres naturels al\351atoires varant de -99 \340 99" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "randmatrix(3,3,antisymmetri c);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Pour une " }{TEXT 338 18 "matrice sym\351trique" }{TEXT -1 54 " avec des nombres naturels al \351atoires varant de 1 \340 10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "randmatrix(4,4,symmetric,entries=rand(-5..10));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Une matrice " }{TEXT 339 23 "triangulaire sup\351rieure" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Triangula ireSup:=matrix(5,5,(i,j)->piecewise(i<=j,3,0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Une matrice " }{TEXT 340 51 "triangulaire inf\351rie ure avec des entr\351es de 20 \340 30" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "rand(a..b)( ) donne un nombre entre a et b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "TriangulaireInf:=matrix(5,5,(i,j)-> piecewise(i>=j,rand(20..30)(),0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 317 10 "Exercices \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 364 6 "NO 1) " }{TEXT -1 8 "Trouvez " }{TEXT 365 2 "a)" }{TEXT -1 5 " F^2 " }{TEXT 366 2 "b) " }{TEXT -1 5 " F^3 " }{TEXT 367 2 "c)" }{TEXT -1 65 " F^n, o\371 F e st la matrice diagonale suivante : F:=diag(1,2,3,5);" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 347 1 " " }{TEXT 329 31 "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 368 6 "NO 2) " } {TEXT -1 62 "Pour quelles valeurs de x,y,z,w a-t-on xA + yB + zC + wF \+ = E ?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "A:= matrix(2,2,[1,3,-5, 3]);B:= matrix(2,2,[9,0,-3,1]);C:= matrix(2,2,[4,2,0,8]);F:= matrix(2, 2,[-4,-6,8,10]);E:= matrix(2,2,[4,5,8,9]);" }{MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 348 1 " " }{TEXT 318 31 "Espace de trava il 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 369 5 "No 3)" } {TEXT 370 1 " " }{TEXT -1 9 "Soit A:= " }{XPPEDIT 18 0 "matrix([[1, 2, 3], [1, 2, 3], [1, 5, 6]]);" "6#-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%F(F )F*7%F(\"\"&\"\"'" }}{PARA 0 "" 0 "" {TEXT -1 24 "R\351soudre Ax = 9x \+ o\371 x = " }{XPPEDIT 18 0 "matrix([[x1], [x2], [x3]])" "6#-%'matrixG6 #7%7#%#x1G7#%#x2G7#%#x3G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 349 1 " " }{TEXT 328 31 "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 -1 0 "" }{TEXT 371 5 "NO 4)" }{TEXT -1 107 " Une matrice A est idempotente si A^2 = A. Pa rmi les matrices suivantes, trouver celle qui est idempotente." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "G:=matrix(3,3,[3,4,7,-3,7,- 12,9,9,8]);H:=matrix([[-25, -75, -125], [7, 21, 35], [1, 3, 5]]);J:=ma trix(3,3,(i,j)->i+j);" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 372 1 " " } {TEXT 327 31 "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 -1 0 "" }{TEXT 373 5 "NO 5)" }{TEXT -1 32 " Trouvez, s'il y a compatibilit\351" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "A:= matrix(3,3,[2,44,-3,8,-2,-80,12,3,10]):B:=randmatrix(4,4):C:=matrix(4, 3,[5,5,6,6,7,7,8,8,9,9,10,10]):E:=matrix(3,3,(i,j)->i-j):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A=evalm(A),B=evalm(B),C=evalm(C),E= evalm(E);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 374 2 "a)" }{TEXT -1 34 " A E et EA. Comparez les r\351ponses\011\n" }{TEXT 375 3 "b) " }{TEXT -1 5 "(BC)E" }{TEXT 376 1 " " }{TEXT -1 30 "et B(CE) Comparez les r\351po nses" }{TEXT 377 5 ". \nc)" }{TEXT -1 8 " (A+E)^2" }{TEXT 378 3 "\nd) " }{TEXT -1 51 " A^2 + 2AE + EA + E^2. Comparez votre r\351ponse avec \+ " }{TEXT 379 2 "c)" }}{PARA 0 "" 0 "" {TEXT 381 2 "e)" }{TEXT -1 49 " \+ A^2 + AE + EA + E^2 Comparez votre r\351ponse avec " }{TEXT 380 2 "c) " }{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 350 1 " " }{TEXT 319 31 "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 -1 0 "" }{TEXT 382 5 "NO 6)" }{TEXT -1 255 " Une matrice A est n ilpotente s'il existe un nombre naturel p tel que A^p = 0 (matrice nul le). La plus petite valeur de p pour laquelle A^p = 0 est appel\351 l' indice de nilpotence. Montrer que la matrice A est nilpotente et trouv er son indice de nilpotence." }}{PARA 0 "" 0 "" {TEXT -1 96 "A=matrix( [[0, 1, 12, 4,7], [0, 0, 7, 10,-8], [0, 0, 0, 15,2], [0, 0, 0, 0,-12], [0, 0, 0, 0,0]]);" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 351 1 " " }{TEXT 333 31 "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 -1 0 "" }{TEXT 383 6 "NO 7) " }{TEXT -1 80 "Si M^2 = I alors a-t -on M = I ou M = -I ? \311tudiez ceci avec la matrice suivante:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "M:=(1/3)*matrix(3,3,[1,-2,-2 ,-2,1,-2,-2,-2,1]);" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 352 1 " " } {TEXT 320 31 "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 384 7 "No 8) " }{TEXT -1 81 "Trouver la plus petite valeur de k tel que M^k = I o\371 M est la matrice suivante :" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "M : = matrix([[sqrt(2)/2,-sqrt(2)/2], [sqrt(2)/2, sqrt(2)/2]]);" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 353 1 " " }{TEXT 321 31 "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 385 7 "No 9) " } {TEXT -1 58 "Soit u = (a,b,c,d);et v = (d,c,b,a), des vecteurs-colonne s" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "a) Trouver une matrice diago nale A telle que Au = v" }}{PARA 0 "" 0 "" {TEXT -1 56 "b) Trouver une matrice non diagonale A telle que Au = v " }}}{SECT 1 {PARA 3 "" 0 " " {TEXT 354 1 " " }{TEXT 322 31 "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 386 8 "No 10) " }{TEXT -1 103 "Soit u = \+ (a,b,c,d) et v = (a + c, b + d), des vecteurs-colonnes. Trouver une ma trice A telle que Au = v" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 357 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 387 8 "No 11) " }{TEXT -1 5 "Soit " }{XPPEDIT 18 0 "A = ma trix([[c, 2, -4], [3, a, 6], [2, 7, b]]);" "6#/%\"AG-%'matrixG6#7%7%% \"cG\"\"#,$\"\"%!\"\"7%\"\"$%\"aG\"\"'7%F+\"\"(%\"bG" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "B = matrix([[-2, a, 4], [b, 6, 5], [4, -2, b]]);" "6 #/%\"BG-%'matrixG6#7%7%,$\"\"#!\"\"%\"aG\"\"%7%%\"bG\"\"'\"\"&7%F.,$F+ F,F0" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "C = matrix([[-32, 0, 42], [30 , -48, -26], [-37, 40, 52]]);" "6#/%\"CG-%'matrixG6#7%7%,$\"#K!\"\"\" \"!\"#U7%\"#I,$\"#[F,,$\"#EF,7%,$\"#PF,\"#S\"#_" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Sachant que C = AB, trou ver a, b et c \340 l'aide seulement de produit scalaire de lignes de A avec des colonnes de B ; " }}{PARA 0 "" 0 "" {TEXT -1 36 "C[i,j] = d otprod(row(A,i),col(B,j))" }}{PARA 0 "" 0 "" {TEXT -1 100 "Note : \311 crire sur votre premi\350re ligne de solution : assume(a,real): assume (b,real): assume(c,real):" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 358 32 " E space 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 "" }}}}}}{MARK "2 0 0" 21 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }