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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 260 67 "Laboratoire 5 - l'inver
se d'une matrice et transpos\351e d'une matrice" }}}{EXCHG {PARA 256 "
" 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 261 51 "Principales commandes utili
s\351es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "mat
rix de la biblioth\350que linalg, randmatrix, transpose, inverse, " }}
{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 
256 24 "Transpos\351e d'une matrice" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "M:=randmatrix(2,4);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "B:=transpose(M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
20 "On a M(i,j) = B(j,i)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"M[2,3]=B[3,2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 46 "Matrice sym
\351trique et matrices antisym\351triques" }}{PARA 0 "" 0 "" {TEXT -1 
56 "Une matrice carr\351e A est sym\351trique si transpos\351e(A) = A
" }}{PARA 0 "" 0 "" {TEXT -1 26 "On a alors A(i,j) = A(j,i)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Symetrique:=randmatrix(3,3,s
ymmetric);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Symetrique[
3,2] =Symetrique[2,3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Une mat
rice carr\351e A est antisym\351trique si transpos\351e(A) = -A. On a \+
alors A(i,j) = -A(j,i)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A
ntiSymetrique:=randmatrix(3,3,antisymmetric,entries=rand(-10..10));\n
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "AntiSymetrique[3,2]=-An
tiSymetrique[2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT 257 22 "Inverse d'une matrice " }{TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Une matrice carr\351e A
(nxn) est inversible s'il existe une matrice B(nxn) telle que AB = BA \+
= I(nxn). B est appel\351e la matrice inverse da A et est not\351e A^(
-1). Dans Maple :" }{TEXT 263 11 " inverse(A)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "A:=randmatrix(3,3,entries=rand(0..20));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "V\351rification : A*inverse(A) = I" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "multiply(A,inverse(A));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Certaines matrices carr\351es ne s
ont pas inversibles" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=m
atrix(2,2,[2,7,4,14]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "i
nverse(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Trouver une matrice
 B telle que AB = I " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=
matrix(3,3,[1,-2,0,2,5,-3,9,-6,0]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "B:=matrix(3,3,[x1,x2,x3,x4,x5,x6,x7,x8,x9]);Id:=diag(
1,1,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AB:=multiply(A,B
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Posons AB = I" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(student,equate):equate(AB,Id);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "solve(%,\{x1,x2,x3,x4,x
5,x6,x7,x8,x9\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(%
,op(B));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Cette matrice est la \+
matrice inverse de A . V\351rifions :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "inverse(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "D'autres m\351thodes ex
istent  pour chercher l'inverse d'une matrice (avec les matrices \351l
\351mentaires, Gauss-Jordan). Voir lab 7 M\351thode GaussJordan" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT 258 9 "Exercices" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 271 5 "No 1)" }{TEXT -1 32 " Trouvez, s'il y a \+
compatibilit\351" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A:=matr
ix(3,2,[1,2,3,4,5,6]);B:=matrix(2,3,[3,6,9,12,15,18]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 273 2 "a)" }{TEXT -1 89 "  transpos\351e(AB)    \+
                Remarque : transpos\351e(M) signifie la transpos\351e \+
de M\n" }{TEXT 272 3 "b) " }{TEXT -1 60 " transpos\351e(A)* transpos
\351e(B)   et comparez la r\351ponse avec " }{TEXT 274 2 "a)" }}{PARA 
0 "" 0 "" {TEXT 275 3 "c) " }{TEXT -1 61 "  transpos\351e(B)* transpos
\351e(A)   et comparez la r\351ponse avec " }{TEXT 276 2 "a)" }{TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 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 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 277 6 "No 2) " }{TEXT -1 
41 "Cr\351er une matrice N, 3x4 avec la commande" }{TEXT 278 13 " rand
matrix. " }{TEXT 279 9 "Trouver :" }{TEXT -1 21 "\na) N*transpos\351e(
N)  " }}{PARA 0 "" 0 "" {TEXT -1 18 "b) transpos\351e(N)*N" }}{PARA 0 
"" 0 "" {TEXT -1 20 "Que remarquez-vous ?" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT 264 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 -1 0 "" }{TEXT 280 6 "No 3) " }{TEXT -1 41 "Cr\351er une ma
trice N, 3x3 avec la commande" }{TEXT 281 13 " randmatrix. " }{TEXT 
282 9 "Trouver :" }{TEXT -1 78 "\na) X = N + transpos\351e(N)). Quelle
 propri\351t\351 remarquez-vous pour la matrice X?" }}{PARA 0 "" 0 "" 
{TEXT -1 78 "b) Y = N - transpos\351e(N)). Quelle propri\351t\351 rema
rquez-vous pour la matrice Y? " }}{PARA 0 "" 0 "" {TEXT -1 193 "c) En \+
conclure que toute matrice carr\351e N est la somme d'une matrice sym
\351trique et d'une matrice antisym\351trique. \311crire N comme la so
mme d'une matrice sym\351trique et d'une matrice antisym\351trique" }}
}{SECT 1 {PARA 3 "" 0 "" {TEXT 265 32 " Espace de travail de l'\351tud
iant" }}{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 283 6 "No 4) " }{TEXT -1 9 "Soit B
 = " }{XPPEDIT 18 0 "matrix([[x, y], [z, w]]);" "6#-%'matrixG6#7$7$%\"
xG%\"yG7$%\"zG%\"wG" }{TEXT -1 88 ", Dans chacun des cas suivants, r
\351soudre le syst\350me d'\351quations g\351n\351r\351 par AB = I(2x2
) :" }}{PARA 0 "" 0 "" {TEXT 285 2 "a)" }{TEXT -1 5 " A = " }{XPPEDIT 
18 0 "matrix([[1, 5], [2, -7]]);" "6#-%'matrixG6#7$7$\"\"\"\"\"&7$\"\"
#,$\"\"(!\"\"" }}{PARA 0 "" 0 "" {TEXT 284 2 "b)" }{TEXT -1 5 " A = " 
}{XPPEDIT 18 0 "matrix([[1, 2], [2, 4]]);" "6#-%'matrixG6#7$7$\"\"\"\"
\"#7$F)\"\"%" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 266 32 " Espace de trav
ail 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 286 6 "No 5) " }
{TEXT -1 9 "Soit B = " }{XPPEDIT 18 0 "matrix([[x, y], [z, w]]);" "6#-
%'matrixG6#7$7$%\"xG%\"yG7$%\"zG%\"wG" }{TEXT -1 8 " et A = " }
{XPPEDIT 18 0 "matrix([[-2, 7], [12, 9]]);" "6#-%'matrixG6#7$7$,$\"\"#
!\"\"\"\"(7$\"#7\"\"*" }{TEXT -1 2 ". " }{TEXT 288 0 "" }}{PARA 0 "" 
0 "" {TEXT 287 3 "a) " }{TEXT -1 68 "Trouver, si possible, une matrice
 B non nulle, telle que AB = 0(2x2)" }}{PARA 0 "" 0 "" {TEXT 289 2 "b)
" }{TEXT -1 54 " V\351rifier si A est inversible avec la commande inve
rse" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 0 "" }{TEXT 267 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 290 6 "No \+
6) " }{TEXT -1 5 "Soit " }{XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%%\"aG%
\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG%\"hG%\"iG" }{TEXT -1 9 " et  A = " }
{XPPEDIT 18 0 "matrix([[2, 3, -5], [-3, 6, 0], [-3, -8, 10]]);" "6#-%'
matrixG6#7%7%\"\"#\"\"$,$\"\"&!\"\"7%,$F)F,\"\"'\"\"!7%,$F)F,,$\"\")F,
\"#5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 2 "a
)" }{TEXT -1 69 " Trouver, si possible, une matrices B non nulle telle
 que AB = 0(2x2)" }}{PARA 0 "" 0 "" {TEXT 291 3 "b) " }{TEXT -1 53 "V
\351rifier si A est inversible avec la commande inverse" }}{PARA 0 "" 
0 "" {TEXT 293 3 "c) " }{TEXT -1 64 "Du no 5) et 6), que conclure des \+
solutions de l'\351quation AB = 0?" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
270 0 "" }{TEXT 269 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 295 5 "No 7)" }{TEXT -1 48 " Une matrice \+
A est d\351finie par A(i,j) = sin(i-j)" }}{PARA 0 "" 0 "" {TEXT -1 79 
"a) Dire si la matrice A est sym\351trique ou antisym\351trique ou ni \+
l'un ni l'autre." }}{PARA 0 "" 0 "" {TEXT -1 62 "b) Construire une mat
rice A(4x4) pour v\351rifier votre r\351ponse. " }}}{SECT 1 {PARA 3 "
" 0 "" {TEXT 294 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 297 5 "No 8)" }{TEXT -1 55 " M\352mes que
stions qu'au num\351ro 7) avec A(i,j) = cos(i-j)" }}{PARA 0 "" 0 "" 
{TEXT -1 79 "a) Dire si la matrice A est sym\351trique ou antisym\351t
rique ou ni l'un ni l'autre." }}{PARA 0 "" 0 "" {TEXT -1 62 "b) Constr
uire une matrice A(4x4) pour v\351rifier votre r\351ponse. " }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT 296 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 299 6 "No 9) " }{TEXT 300 44 "G\351n\351r
er 2 matrices 4x4, A et B, sym\351triques." }}{PARA 0 "" 0 "" {TEXT 
301 29 "a) A + B est-elle sym\351trique?" }}{PARA 0 "" 0 "" {TEXT -1 
11 "b) 3A - 4B " }{TEXT 304 20 "est-elle sym\351trique?" }}{PARA 0 "" 
0 "" {TEXT 302 26 "c) AB est-elle sym\351trique?" }}{PARA 0 "" 0 "" 
{TEXT 303 78 "d) Recommencer avec d'autres matrices A et B. Que vous s
ugg\350rent les r\351ponses?" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 298 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 306 6 "No 10)" }{TEXT 308 58 " Trouver 2 matrices 4x4, sym\351tr
iques telles que le produit" }{TEXT 307 1 " " }{TEXT 309 18 "AB est sy
m\351trique?" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 305 32 " Espace de trav
ail 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 -1 216 "* 11) Soi
t A une matrice carr\351e nxn. L'\351l\351ment sym\351trique centralem
ent \340 l'\351l\351ment A[i,j] est l'\351l\351ment qui lui est sym
\351trique par rapport au centre de la matrice, math\351matiquement, c
'est l'\351l\351ment A(n-i+1, n-j+1).  " }}{PARA 0 "" 0 "" {TEXT -1 
176 "Une matrice poss\350de une sym\351trie centrale(SC) si A[i,j]  = \+
A(n-i+1, n-j+1) et une matrice poss\350de une antisym\351trie centrale
 (ASC) si A[i,j]  = - A(n-i+1, n-j+1). Exemple : B = " }{XPPEDIT 18 0 
"matrix([[134, 111, 116], [85, 194, 85], [116, 111, 134]])" "6#-%'matr
ixG6#7%7%\"$M\"\"$6\"\"$;\"7%\"#&)\"$%>F,7%F*F)F(" }{TEXT -1 17 " a un
e SC et C = " }{XPPEDIT 18 0 "matrix([[36, -1, -42], [-15, 0, 15], [42
, 1, -36]])" "6#-%'matrixG6#7%7%\"#O,$\"\"\"!\"\",$\"#UF+7%,$\"#:F+\"
\"!F07%F-F*,$F(F+" }{TEXT -1 10 " a une ASC" }}{PARA 0 "" 0 "" {TEXT 
-1 65 "a) Montrer que toute matrice A = B + C o\371 B a une SC et B un
e ASC" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exemple :  " }{XPPEDIT 18 0 "mat
rix([[170, 110, 74], [70, 194, 100], [158, 112, 98]]);" "6#-%'matrixG6
#7%7%\"$q\"\"$5\"\"#u7%\"#q\"$%>\"$+\"7%\"$e\"\"$7\"\"#)*" }{TEXT -1 
3 " = " }{XPPEDIT 18 0 "matrix([[134, 111, 116], [85, 194, 85], [116, \+
111, 134]]);" "6#-%'matrixG6#7%7%\"$M\"\"$6\"\"$;\"7%\"#&)\"$%>F,7%F*F
)F(" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "matrix([[36, -1, -42], [-15, 0,
 15], [42, 1, -36]]);" "6#-%'matrixG6#7%7%\"#O,$\"\"\"!\"\",$\"#UF+7%,
$\"#:F+\"\"!F07%F-F*,$F(F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 23 "b) Soit la matrice A = " }{XPPEDIT 18 0 "matrix([
[1, 5, 2, 8], [3, 7, -5, -2], [4, 1, 11, 1], [12, 8, -3, -1]]);" "6#-%
'matrixG6#7&7&\"\"\"\"\"&\"\"#\"\")7&\"\"$\"\"(,$F)!\"\",$F*F07&\"\"%F
(\"#6F(7&\"#7F+,$F-F0,$F(F0" }{TEXT -1 63 " Trouver B et C o\371 A = B
 + C telle que B a une SC et C une ASC " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 310 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 "" }}}}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }