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:Z:j;<Z:>:1:" }{TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 257 14 "Pivot de
 Gauss" }}{PARA 19 "" 0 "" {TEXT 259 40 "\251 Apprendre Maple. Alain L
e Stang. 2006\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(lina
lg);" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm
 and trace have been redefined and unprotected\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco
mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(
augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch
arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co
ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro
wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv
ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg
ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG
%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG
%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%
'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG
%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'
mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG
%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%
'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith
G%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG
%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%
(vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 
11 "Question 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "Permute
Lignes:=proc(A::matrix,l1::posint,l2::posint) \nprint(`Echange des lig
nes`,L[l1],L[l2]);                 A:=swaprow(A,l1,l2); \nend proc;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.PermuteLignesGf*6%'%\"AG%'matrixG'
%#l1G%'posintG'%#l2GF,6\"F/F/C$-%&printG6%%3Echange~des~lignesG&%\"LG6
#9%&F66#9&>9$-%(swaprowG6%F=F8F;F/F/F/" }}}{EXCHG {PARA 18 "" 0 "" 
{TEXT -1 11 "Question 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
197 "PermuteColonnes:=proc(A::matrix,c1::posint,c2::posint,T::array)\n
  local temp;\n  temp:=T[c1];T[c1]:=T[c2];T[c2]:=temp;\n  print(`Echan
ge des colonnes `,C[c1],C[c2]); \n  A:=swapcol(A,c1,c2);\nend proc;" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0PermuteColonnesGf*6&'%\"AG%'matrix
G'%#c1G%'posintG'%#c2GF,'%\"TG%&arrayG6#%%tempG6\"F4C'>8$&9'6#9%>F8&F9
6#9&>F=F7-%&printG6%%6Echange~des~colonnes~G&%\"CGF:&FFF>>9$-%(swapcol
G6%FIF;F?F4F4F4" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Question 3:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "CherchePivotLigne:=proc(
A::matrix,l1::posint,l2::posint,c::posint)\n  local i,ligne,maxi;\n  l
igne:=0;maxi:=0;\n  for i from l1 to l2 do\n    if A[i,c]<>0 and abs(A
[i,c])>maxi then\n      maxi:=abs(A[i,c]):ligne:=i\n    end if;\n  end
 do;\n  ligne;\n  end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2Cher
chePivotLigneGf*6&'%\"AG%'matrixG'%#l1G%'posintG'%#l2GF,'%\"cGF,6%%\"i
G%&ligneG%%maxiG6\"F5C&>8%\"\"!>8&F9?(8$9%\"\"\"9&%%trueG@$30&9$6$F=9'
F92F;-%$absG6#FEC$>F;FJ>F8F=F8F5F5F5" }}}{EXCHG {PARA 18 "" 0 "" 
{TEXT -1 11 "Question 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
248 "CherchePivotColonne:=proc(A::matrix,c1::posint,c2::posint,l::posi
nt)\nlocal j,colonne,maxi;\ncolonne:=0:maxi:=0:\nfor j from c1 to c2 d
o  \n  if A[l,j]<>0 and abs(A[l,j])>maxi then\n    maxi:=abs(A[l,j]);c
olonne:=j\n  end if;\nend do;\ncolonne;\nend proc;  " }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%4CherchePivotColonneGf*6&'%\"AG%'matrixG'%#c1G%'po
sintG'%#c2GF,'%\"lGF,6%%\"jG%(colonneG%%maxiG6\"F5C&>8%\"\"!>8&F9?(8$9
%\"\"\"9&%%trueG@$30&9$6$9'F=F92F;-%$absG6#FEC$>F;FJ>F8F=F8F5F5F5" }}}
{EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Question 5:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 128 "AfficheLigne:=proc(A::matrix,i::posint,T::ar
ray)\nlocal j,p,X;\np:=coldim(A);\nprint(sum(A[i,j]*X[T[j]],j=1..p-1)=
A[i,p])\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-AfficheLigneG
f*6%'%\"AG%'matrixG'%\"iG%'posintG'%\"TG%&arrayG6%%\"jG%\"pG%\"XG6\"F4
C$>8%-%'coldimG6#9$-%&printG6#/-%$sumG6$*&&F;6$9%8$\"\"\"&8&6#&9&6#FGF
H/FG;FH,&F7FHFH!\"\"&F;6$FFF7F4F4F4" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 
-1 14 "Question 6 a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "A
fficheSolutions1:=proc(A::matrix,T::array)\nlocal i,k,n,x,X,d; \nn:=ro
wdim(A);\nfor i from n by -1 to 1 do\n  d:=0;\n  for k from 1+i to n d
o d:=d+A[i,k]*x[T[k]] end do;\n  x[T[i]]:=A[i,n+1]-d; \n  end do;\n  p
rint(`Solution unique :`);\n  for i to n do print(X[T[i]]=x[T[i]]) end
 do; \nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2AfficheSolution
s1Gf*6$'%\"AG%'matrixG'%\"TG%&arrayG6(%\"iG%\"kG%\"nG%\"xG%\"XG%\"dG6
\"F4C&>8&-%'rowdimG6#9$?(8$F7!\"\"\"\"\"%%trueGC%>8)\"\"!?(8%,&F=F?F?F
?F?F7F@>FC,&FCF?*&&F;6$F=FFF?&8'6#&9%6#FFF?F?>&FN6#&FQ6#F=,&&F;6$F=,&F
7F?F?F?F?FCF>-%&printG6#%2Solution~unique~:G?(F=F?F?F7F@-Fgn6#/&8(FUFT
F4F4F4" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 14 "Question 6 b):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "AfficheSolutions2:=proc(A::
matrix,r::nonnegint,T::array) \nlocal X,i,k,m,n,x,d;\nm:=0;n:=rowdim(A
);\nfor i from r+1 to n do\n  if A[i,r+1]<>0 then m:=m+1 end if;\nend \+
do;\nif m>0 then print(`Pas de solution.`) \nelse \n  for i from r by \+
-1 to 1 do\n    d:=0;\n    for k from 1+i to r do d:=d+A[i,k]*x[T[k]] \+
end do;\n    x[T[i]]:=A[i,r+1]-d;    \n  end do; \n  print(`Solution u
nique :`);\n  for i to r do print(X[T[i]]=x[T[i]]) end do;\nend if;\ne
nd proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2AfficheSolutions2Gf*6%'
%\"AG%'matrixG'%\"rG%*nonnegintG'%\"TG%&arrayG6)%\"XG%\"iG%\"kG%\"mG%
\"nG%\"xG%\"dG6\"F8C&>8'\"\"!>8(-%'rowdimG6#9$?(8%,&9%\"\"\"FGFGFGF>%%
trueG@$0&FB6$FDFEF<>F;,&F;FGFGFG@%2F<F;-%&printG6#%1Pas~de~solution.GC
%?(FDFF!\"\"FGFHC%>8*F<?(8&,&FDFGFGFGFGFFFH>FZ,&FZFG*&&FB6$FDFfnFG&8)6
#&9&6#FfnFGFG>&F^o6#&Fao6#FD,&FKFGFZFW-FR6#%2Solution~unique~:G?(FDFGF
GFFFH-FR6#/&8$FeoFdoF8F8F8" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 14 "Qu
estion 6 c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 527 "AfficheSol
utions3:=proc(A::matrix,r::nonnegint,T::array)\nlocal i,k,m,n,p,x,X,w,
d,t,B;\nn:=rowdim(A);p:=coldim(A)-1;\nm:=0;\nfor i from r+1 to n do\n \+
if A[i,p+1]<>0 then m:=m+1 end if;\nend do; \nif m>0 then RETURN(`Pas \+
de solution.`) \nelse\n  for i from r by -1 to 1 do\n    d:=0;\n    fo
r k from 1+i to p do d:=d+A[i,k]*X[T[k]] end do;\n    X[T[i]]:=A[i,p+1
]-d;    \n  end do; \n  for k from 1 to r do\n    print(` X`[T[k]]=X[T
[k]]);\n  end do;  \n  for k from r+1 to p do \n    print(X[T[k]],` r
\351el quelconque`) \n  end do;\nend if;\nend proc; " }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%2AfficheSolutions3Gf*6%'%\"AG%'matrixG'%\"rG%*nonn
egintG'%\"TG%&arrayG6-%\"iG%\"kG%\"mG%\"nG%\"pG%\"xG%\"XG%\"wG%\"dG%\"
tG%\"BG6\"F<C'>8'-%'rowdimG6#9$>8(,&-%'coldimGFB\"\"\"FI!\"\">8&\"\"!?
(8$,&9%FIFIFIFIF?%%trueG@$0&FC6$FO,&FEFIFIFIFM>FL,&FLFIFIFI@%2FMFL-%'R
ETURNG6#%1Pas~de~solution.GC%?(FOFQFJFIFRC%>8,FM?(8%,&FOFIFIFIFIFEFR>F
^o,&F^oFI*&&FC6$FOF`oFI&8*6#&9&6#F`oFIFI>&Fho6#&F[p6#FO,&FUFIF^oFJ?(F`
oFIFIFQFR-%&printG6#/&%#~XGFioFgo?(F`oFPFIFEFR-Fep6$Fgo%1~r|dyel~quelc
onqueGF<F<F<" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Question 7:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1569 "PivotGauss:=proc(A::matrix
,rg)\nlocal n,p,code,d,T,pivot,i,j,k,l,k1,k2,r;\n\nn:=rowdim(A);p:=col
dim(A)-1;\nT:=array(1..p);\nfor i to p do T[i]:=i end do;\nfor i to n \+
do AfficheLigne(A,i,T) end do;\nprint(`-------------------------------
-------`);  \ni:=0;r:=0;\nwhile i<min(n,p) do\n  i:=i+1;\n  pivot:=A[i
,i];\n  code:=NULL;\n  if pivot<>0 then\n    r:=r+1;\n    for k to p+1
 do A[i,k]:=A[i,k]/pivot end do;\n    if pivot<>1 then \n       code:=
code,L[i]=L[i]/pivot\n    end if; \n    for l from i+1 to n do\n      \+
 d:=A[l,i];\n       A:=addrow(A,i,l,-d);\n       if d<>0 then\n       \+
    code:=code,L[l]=L[l]-L[i]*d/pivot end if;\n    end do;  \n    for \+
k to nops([code]) do\n         print(op(k,[code])) \n    end do;\n    \+
print(` `); \n    for k to n do AfficheLigne(A,k,T) end do;\n  print(`
--------------------------------------`);                        \n   \+
 else   # si A[i,i]=0\n        \n    k1:=CherchePivotLigne(A,i+1,n,i);
   \n    if k1>0 then  \n       PermuteLignes(A,i,k1);i:=i-1; \n    el
se                        \n      k2:=CherchePivotColonne(A,i+1,p,i);
\n      if k2>0 then \n         PermuteColonnes(A,i,k2,T);i:=i-1; \n  \+
    else\n      for j from p to i+1 by -1 do   \n           k:=Cherche
PivotLigne(A,i+1,n,j);\n           if k>0 then\n             PermuteLi
gnes(A,i,k);\n             PermuteColonnes(A,i,j,T);\n             i:=
i-1;break; \n           end if;\n      end do;\n     end if;\n    end \+
if;\n   end if;\n \nend do; \nrg:=r;\nif r=n and r=p then AfficheSolut
ions1(A,T) end if; \nif r=p and r<n then AfficheSolutions2(A,r,T) end \+
if; \nif r<p then AfficheSolutions3(A,r,T) end if;\nend proc; " }}
{PARA 12 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+Pi
votGaussGf*6$'%\"AG%'matrixG%#rgG6/%\"nG%\"pG%%codeG%\"dG%\"TG%&pivotG
%\"iG%\"jG%\"kG%\"lG%#k1G%#k2G%\"rG6\"F9C/>8$-%'rowdimG6#9$>8%,&-%'col
dimGF?\"\"\"FF!\"\">8(-%&arrayG6#;FFFB?(8*FFFFFB%%trueG>&FI6#FOFO?(FOF
FFFF<FP-%-AfficheLigneG6%F@FOFI-%&printG6#%G--------------------------
------------G>FO\"\"!>80Fgn?(F9FFFFF92FO-%$minG6$F<FBC&>FO,&FOFFFFFF>8
)&F@6$FOFO>8&%%NULLG@%0FcoFgnC*>Fin,&FinFFFFFF?(8,FFFF,&FBFFFFFFFP>&F@
6$FOF_p*&FbpFFFcoFG@$0FcoFF>Fgo6$Fgo/&%\"LGFS*&FjpFFFcoFG?(8-FaoFFF<FP
C%>8'&F@6$F^qFO>F@-%'addrowG6&F@FOF^q,$FaqFG@$0FaqFgn>Fgo6$Fgo/&F[q6#F
^q,&F^rFF*(FjpFFFaqFFFcoFGFG?(F_pFFFF-%%nopsG6#7#FgoFP-FY6#-%#opG6$F_p
Ffr-FY6#%\"~G?(F_pFFFFF<FP-FV6%F@F_pFIFXC$>8.-%2CherchePivotLigneG6&F@
FaoF<FO@%2FgnFdsC$-%.PermuteLignesG6%F@FOFds>FO,&FOFFFFFGC$>8/-%4Cherc
hePivotColonneG6&F@FaoFBFO@%2FgnFbtC$-%0PermuteColonnesG6&F@FOFbtFI>FO
F_t?(8+FBFGFaoFPC$>F_p-Ffs6&F@FaoF<F^u@$2FgnF_pC&-F\\t6%F@FOF_p-Fjt6&F
@FOF^uFI>FOF_t[>9%Fin@$3/FinF</FinFB-%2AfficheSolutions1G6$F@FI@$3Fav2
FinF<-%2AfficheSolutions2G6%F@FinFI@$2FinFB-%2AfficheSolutions3GFjvF9F
9F9" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 14 "Question 8 a):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M:=matrix([[2,-1,3,9],[3,1,-
1,7],[5,-2,1,5]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG
6#7%7&\"\"#!\"\"\"\"$\"\"*7&F,\"\"\"F+\"\"(7&\"\"&!\"#F/F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "PivotGauss(M,'r');print(`rang `=r);
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"\"\"\"#&F&6#F)!\"
\"*&\"\"$F(&F&6#F.F(F(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"
XG6#\"\"\"\"\"$&F&6#\"\"#F(&F&6#F)!\"\"\"\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(&%\"XG6#\"\"\"\"\"&*&\"\"#F(&F&6#F+F(!\"\"&F&6#\"\"$
F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G----------------------------
----------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"\",$F$#F'
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"#,&F$\"\"\"*&#\"
\"$F'F)&F%6#F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"
$,&F$\"\"\"*&#\"\"&\"\"#F)&F%6#F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"\"F(*&#F
(\"\"#F(&F&6#F+F(!\"\"*&#\"\"$F+F(&F&6#F1F(F(#\"\"*F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"##\"\"&F(*&#\"#6F(\"\"\"&F&6#\"\"$F
.!\"\"#!#8F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"##\"\"
\"F(*&#\"#8F(F*&F&6#\"\"$F*!\"\"#!#NF(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"LG6#\"\"#,$F$#F'\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"LG6#\"\"$,&F$\"\"\"*&#F)\"\"&F)&F%6#\"\"#F)!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,(&%\"XG6#\"\"\"F(*&#F(\"\"#F(&F&6#F+F(!\"\"*&#\"\"$F+F(&F&6#F1
F(F(#\"\"*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"#\"\"\"
*&#\"#6\"\"&F)&F&6#\"\"$F)!\"\"#!#8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/,$&%\"XG6#\"\"$#!#F\"\"&#!#\")F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%G--------------------------------------G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"LG6#\"\"$,$F$#!\"&\"#F" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"
\"\"F(*&#F(\"\"#F(&F&6#F+F(!\"\"*&#\"\"$F+F(&F&6#F1F(F(#\"\"*F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"#\"\"\"*&#\"#6\"\"&F)&F
&6#\"\"$F)!\"\"#!#8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#\"\"
$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G-----------------------------
---------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Solution~unique~:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#\"\"\"\"\"#" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/&%\"XG6#\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/&%\"XG6#\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&rang~G\"\"$
" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 14 "Question 8 b):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "M:=matrix([[2,-1,1,5],[1,1,-2,-2],[
4,1,-3,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7&
\"\"#!\"\"\"\"\"\"\"&7&F,F,!\"#F/7&\"\"%F,!\"$F," }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 35 "PivotGauss(M,'r');print(`rang `=r);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"\"\"\"#&F&6#F)!\"\"&F&6#\"\"$
F(\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"\"F(&F&6#\"
\"#F(*&F+F(&F&6#\"\"$F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,
(&%\"XG6#\"\"\"\"\"%&F&6#\"\"#F(*&\"\"$F(&F&6#F.F(!\"\"F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"\",$F$#F'\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"#,&F$\"\"\"*&#F)F'F)&F%6#F)F)!
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"$,&F$\"\"\"*&\"\"
#F)&F%6#F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"\"F(*&#F(\"\"#F(&F&6#F+F(!\"
\"*&#F(F+F(&F&6#\"\"$F(F(#\"\"&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
,&&%\"XG6#\"\"##\"\"$F(*&#\"\"&F(\"\"\"&F&6#F*F.!\"\"#!\"*F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"#\"\"$*&\"\"&\"\"\"&F&6#F)F,!
\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G------------------------
--------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"#,$F$#
F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"$,&F$\"\"\"*&
\"\"#F)&F%6#F+F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"\"F(*&#F(\"\"#F(&F&6#F+
F(!\"\"*&#F(F+F(&F&6#\"\"$F(F(#\"\"&F+" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&&%\"XG6#\"\"#\"\"\"*&#\"\"&\"\"$F)&F&6#F-F)!\"\"!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%#~XG6#\"\"\",&F'F'*&#F'\"\"$F'&%\"XG6#F+F'F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#~XG6#\"\"#,&!\"$\"\"\"*&#\"\"&\"\"
$F*&%\"XG6#F.F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"XG6#\"\"$%1~r
|dyel~quelconqueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&rang~G\"\"#" }
}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 14 "Question 8 c):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "M:=matrix([[1,1,1,1,4,5],[1,-1,1,-1
,0,-2],[2,0,2,1,5,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'ma
trixG6#7%7(\"\"\"F*F*F*\"\"%\"\"&7(F*!\"\"F*F.\"\"!!\"#7(\"\"#F/F2F*F,
F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "PivotGauss(M,'r');pri
nt(`rang `=r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"XG6#\"\"\"F(&
F&6#\"\"#F(&F&6#\"\"$F(&F&6#\"\"%F(*&F1F(&F&6#\"\"&F(F(F5" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/,*&%\"XG6#\"\"\"F(&F&6#\"\"#!\"\"&F&6#\"\"$F(&
F&6#\"\"%F,!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"XG6#\"\"\"\"
\"#*&F)F(&F&6#\"\"$F(F(&F&6#\"\"%F(*&\"\"&F(&F&6#F2F(F(F0" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"#,&F$\"\"\"&F%6#F)!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"LG6#\"\"$,&F$\"\"\"*&\"\"#F)&F%
6#F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/,,&%\"XG6#\"\"\"F(&F&6#\"\"#F(&F&6#\"\"$F(&F&6#\"\"
%F(*&F1F(&F&6#\"\"&F(F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG
6#\"\"#!\"#*&F(\"\"\"&F&6#\"\"%F+!\"\"*&F.F+&F&6#\"\"&F+F/!\"(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"XG6#\"\"#!\"#&F&6#\"\"%!\"\"*&
\"\"$\"\"\"&F&6#\"\"&F0F-!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G---
-----------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/&%\"LG6#\"\"#,$F$#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"L
G6#\"\"$,&F$\"\"\"&F%6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"XG6#\"\"\"F(&F&6#\"\"#F
(&F&6#\"\"$F(&F&6#\"\"%F(*&F1F(&F&6#\"\"&F(F(F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(&%\"XG6#\"\"#\"\"\"&F&6#\"\"%F)*&F(F)&F&6#\"\"&F)F)#
\"\"(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"%\"\"\"&F&6#
\"\"&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G-----------------------
---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%6Echange~des~colo
nnes~G&%\"CG6#\"\"$&F%6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"XG6#\"\"\"F(&F&6#\"\"#F(&F&
6#\"\"%F(&F&6#\"\"$F(*&F.F(&F&6#\"\"&F(F(F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(&%\"XG6#\"\"#\"\"\"&F&6#\"\"%F)*&F(F)&F&6#\"\"&F)F)#
\"\"(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"XG6#\"\"%\"\"\"&F&6#
\"\"&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G-----------------------
---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#~XG6#\"\"\",(#
\"\"$\"\"#F'*&F+F'&%\"XG6#\"\"&F'!\"\"&F.6#F*F1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%#~XG6#\"\"#,&#\"\"&F'\"\"\"&%\"XG6#F*!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%#~XG6#\"\"%,&\"\"\"F)&%\"XG6#\"\"&!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"XG6#\"\"$%1~r|dyel~quelconqueG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%\"XG6#\"\"&%1~r|dyel~quelconqueG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&rang~G\"\"$" }}}}{MARK "2 0" 40 }
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