%% Created by Maple V Release 4 (IBM INTEL NT)
%% Source Worksheet: TD9corrMat.mws
%% Generated: Sun Apr 29 00:26:00 2007
\documentclass{article}
\usepackage{maple2e}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
\begin{document}
\begin{maplegroup}
NOYAU et IMAGE

\mapleresult
\begin{maplettyout}
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{A:=matrix(2,2,[-1/5,-2/5,3/5,6/5]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
A :=  \left[ 
{\begin{array}{cc}
{\displaystyle \frac {-1}{5}}  & {\displaystyle \frac {-2}{5}} 
 \\ [2ex]
{\displaystyle \frac {3}{5}}  & {\displaystyle \frac {6}{5}} 
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalm(A*A);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
 \left[ 
{\begin{array}{cc}
{\displaystyle \frac {-1}{5}}  & {\displaystyle \frac {-2}{5}} 
 \\ [2ex]
{\displaystyle \frac {3}{5}}  & {\displaystyle \frac {6}{5}} 
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{kernel(A);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\{[-2, \,1]\}
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{colspace(A);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\{[1, \,-3]\}
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
PUISSANCES DE MATRICES

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{A:=(1/3)*matrix(3,3,[0,-2,-2,2,0,-1,2,1,0]);}
{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
A := {\displaystyle \frac {1}{3}} \, \left[ 
{\begin{array}{rrr}
0 & -2 & -2 \\
2 & 0 & -1 \\
2 & 1 & 0
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{seq(evalm(A^k),k=2..5);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
 \left[ 
{\begin{array}{ccc}
{\displaystyle \frac {-8}{9}}  & {\displaystyle \frac {-2}{9}} 
 & {\displaystyle \frac {2}{9}}  \\ [2ex]
{\displaystyle \frac {-2}{9}}  & {\displaystyle \frac {-5}{9}} 
 & {\displaystyle \frac {-4}{9}}  \\ [2ex]
{\displaystyle \frac {2}{9}}  & {\displaystyle \frac {-4}{9}}  & 
{\displaystyle \frac {-5}{9}} 
\end{array}}
 \right] , \, \left[ 
{\begin{array}{ccc}
0 & {\displaystyle \frac {2}{3}}  & {\displaystyle \frac {2}{3}} 
 \\ [2ex]
{\displaystyle \frac {-2}{3}}  & 0 & {\displaystyle \frac {1}{3}
}  \\ [2ex]
{\displaystyle \frac {-2}{3}}  & {\displaystyle \frac {-1}{3}} 
 & 0
\end{array}}
 \right] , \, \left[ 
{\begin{array}{ccc}
{\displaystyle \frac {8}{9}}  & {\displaystyle \frac {2}{9}}  & 
{\displaystyle \frac {-2}{9}}  \\ [2ex]
{\displaystyle \frac {2}{9}}  & {\displaystyle \frac {5}{9}}  & 
{\displaystyle \frac {4}{9}}  \\ [2ex]
{\displaystyle \frac {-2}{9}}  & {\displaystyle \frac {4}{9}}  & 
{\displaystyle \frac {5}{9}} 
\end{array}}
 \right] , \, \left[ 
{\begin{array}{ccc}
0 & {\displaystyle \frac {-2}{3}}  & {\displaystyle \frac {-2}{3}
}  \\ [2ex]
{\displaystyle \frac {2}{3}}  & 0 & {\displaystyle \frac {-1}{3}
}  \\ [2ex]
{\displaystyle \frac {2}{3}}  & {\displaystyle \frac {1}{3}}  & 0
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{B:=matrix(2,2,[1,a,0,1]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
B :=  \left[ 
{\begin{array}{rc}
1 & a \\
0 & 1
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{seq(evalm(B^k),k=2..5);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
 \left[ 
{\begin{array}{rc}
1 & 2\,a \\
0 & 1
\end{array}}
 \right] , \, \left[ 
{\begin{array}{rc}
1 & 3\,a \\
0 & 1
\end{array}}
 \right] , \, \left[ 
{\begin{array}{rc}
1 & 4\,a \\
0 & 1
\end{array}}
 \right] , \, \left[ 
{\begin{array}{rc}
1 & 5\,a \\
0 & 1
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{J:=matrix(4,4,[1$16]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
J :=  \left[ 
{\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{seq(evalm(J^k),k=2..5);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
 \left[ 
{\begin{array}{rrrr}
4 & 4 & 4 & 4 \\
4 & 4 & 4 & 4 \\
4 & 4 & 4 & 4 \\
4 & 4 & 4 & 4
\end{array}}
 \right] , \, \left[ 
{\begin{array}{rrrr}
16 & 16 & 16 & 16 \\
16 & 16 & 16 & 16 \\
16 & 16 & 16 & 16 \\
16 & 16 & 16 & 16
\end{array}}
 \right] , \, \left[ 
{\begin{array}{rrrr}
64 & 64 & 64 & 64 \\
64 & 64 & 64 & 64 \\
64 & 64 & 64 & 64 \\
64 & 64 & 64 & 64
\end{array}}
 \right] , \, \left[ 
{\begin{array}{rrrr}
256 & 256 & 256 & 256 \\
256 & 256 & 256 & 256 \\
256 & 256 & 256 & 256 \\
256 & 256 & 256 & 256
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
INVERSE DE MATRICE

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{A:=matrix(3,3,[1,0,a,1,a,-1,a,0,1]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
A :=  \left[ 
{\begin{array}{ccc}
1 & 0 & a \\
1 & a & -1 \\
a & 0 & 1
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{det(A);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
a - a^{3}
\]
\end{maplelatex}

\end{maplegroup}

\begin{maplegroup}
MARKOV

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{P:=matrix(3,3,[0.8,0.3,0.2,0.1,0.2,0.6,0.1,0.
5,0.2]);}{%
}
\end{mapleinput}

\begin{mapleinput}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
P :=  \left[ 
{\begin{array}{ccc}
.8 & .3 & .2 \\
.1 & .2 & .6 \\
.1 & .5 & .2
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{X:=matrix(3,1,[1000,0,0]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
X :=  \left[ 
{\begin{array}{r}
1000 \\
0 \\
0
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalm(P^7&*X);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
 \left[ 
{\begin{array}{c}
563.8874000 \\
226.0691000 \\
210.0435000
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{I3:=matrix(3,3,[1,0,0,0,1,0,0,0,1]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
{\it I3} :=  \left[ 
{\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{O3:=matrix(3,1,[0$3]);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
{\rm O3} :=  \left[ 
{\begin{array}{r}
0 \\
0 \\
0
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{X:=linsolve((P-I3),O3);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
X :=  \left[ 
{\begin{array}{c}
{{{\it \_t}_{1}}_{1}} \\
.4117647058\,{{{\it \_t}_{1}}_{1}} \\
.3823529411\,{{{\it \_t}_{1}}_{1}}
\end{array}}
 \right] 
\]
\end{maplelatex}

\end{maplegroup}

\end{document}
%% End of Maple V Output
