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\begin{document}

\everymath={\displaystyle}

{\bf Pr{\'e}pa CAPES UPMC 2006} \hfill Emmanuel FERRAND, Laurent KOELBLEN.
\bigskip

\centerline{\bf TD d'Alg{\`e}bre du mercredi 29 novembre 2006}

\bigskip




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
R\'esoudre dans $\RR$ les syst\`emes suivants~:
$$
\left\{
\begin{array}{rrrrrrrrl}
3x&+&4y&+&z&+&2t&=& 3 \\
6x&+&6y&+&2z&+&5t&=& 7\\    
 9x&+&12y&+&3z&+&10t&=& 13\\
\end{array}
\right.
\quad
\left\{
\begin{array}{rrrrrrrrl}
2x&+&3y&+&z&+&2t   &=& 4  \\ 4x&+&3y&+&z&+&t   &=&5\\
6x&+&11y&+&3z&+&2t    &=&2\\ x&-&7y&-&z&+&2t&=&7
\end{array}
\right.
$$
$$
\left\{
\begin{array}{rrrrrrrrrcl}
x_{1}&+&x_{2}& & & & & & &=& 2a_{1}\\
& &x_{2}&+&x_{3} & & & & &=& 2a_{2}\\
& & & & &\ddots& & &   &\vdots& \ \ \vdots \\
& & & & & & x_{n-1}&+&x_{n} &=&2 a_{n-1}\\
x_{1}&+& & & & & &+&x_{n}&=&2a_{n}
\end{array}
\right.
$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
R\'esoudre, en discutant suivant le param\`etre r\'eel $m$
les syst\`emes suivants~:
$$
\left\{
\begin{array}{rrrrrrl}
x&+&my&+& \hfil z    &=&1   \\  mx&+&y&+&(m-1)z  &=&m\\
x&+&y&+& \hfil z    &=& m+1\\
\end{array}
\right.
\quad
\left\{
\begin{array}{rrrrrrl}
mx&+&my&+&z   &=&  1 \\ x&+&m^2y&+&z   &=&m\\
x&+&my&+&mz    &=&1\\
\end{array}
\right.
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
Montrer que
$
\left|
\begin{array}{ccc}
x&a & a\\ a & x& b\\ b& b& x\\
\end{array}
\right|
= (x+a+b)(x-a)(x-b)
$
et
$
\left|
\begin{array}{ccc}
1&\cos a & \cos 2a \\ \cos a &\cos 2a & \cos
3a\\ \cos 2a& \cos 3a&\cos 4a \\
\end{array}
\right|
= 0
$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
Que vaut le produit des deux
d\'eterminants
$
\Delta_1=
\left|
\begin{array}{cccc}
a &b & c&d\\ c & d& a&b\\ b & a&
d& c\\ d&c&b&a\\
\end{array}
\right|
$
et
$
\Delta_2=
\left|
\begin{array}{cccc}
1&\ \ 1 &\ \ 1&\ 1\\  1 &\ \ 1 & -1&-1\\
1&-1 &-1 &\ \ 1\\ 1&-1&\ \ 1&-1\\
\end{array}
\right|
$~?

En d\'eduire la valeur de $\Delta_1$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
Les matrices suivantes sont-elles semblables~? \'equivalentes~? 
$$A=
\left(
\begin{array}{cccc}
1 & 1 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 1 \\ 
0 & 0 & 0 & 1 \\
\end{array}
\right)
\quad
B=
\left( 
\begin{array}{cccc}
1 & 1 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\
\end{array}
\right)
\quad
C=\left( 
\begin{array}{cccc}
1 & 1 & 0 & 0 \\ 
0 & 1 & 1 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\
\end{array}
\right)
\quad
D= \left( 
\begin{array}{cccc}
0 & 0 & 0 & 1 \\ 
0 & 0 & 1 & 1 \\ 
0 & 1 & 4 & 0 \\ 
1 & 0 & 0 & 0 \\
\end{array}
\right)
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
Donner la d\'ecomposition $D+N$ de la matrice
$A=
\left( 
\begin{array}{ccc}
1 & 2 & 3 \\ 
0 & 4 & 5 \\ 
0 & 0 & 6 \\
\end{array}
\right)
$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{}
Soient $A$ et $B$ deux ensembles de cardinal $n$ et $p \leq n$.

\begin{enumerate}

\item
Quel est le nombre d'applications de $A$ dans $B$~?

\item
On note $s(n,k)$ le nombre de surjections d'un ensemble \`a $n$
\'el\'ements sur un ensemble \`a $k$ \'el\'ements.
Montrer que $p^{n} = \sum_{k=0}^{p} {p \choose k} s(n,k)$.
(Remarquer qu'une application de $A$ dans $B$ est la donn\'ee d'une
surjection  de $A$ sur une partie de $B$.)

\item
En d\'eduire $s(n,p)$ (Consid\'erer l'endomorphisme
de $\RR^{n+1}$ dont la matrice est celle du syst\`eme
lin\'eaire reliant $(0^n,\ldots,n^n)$ à $\bigl(s(n,0),\ldots,s(n,n)\bigr)$.)
 
\end{enumerate}


\end{document}