% matrices
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%%%     Feuille d'exercices de mathématiques au format LaTeX     %%%
%%%     créée le Fri, 17 Oct 2008 01:41:13 +0200                 %%%
%%%     sur http://allken-bernard.org/pierre/phpmyexercices      %%%
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% Apparence des exercices
\newcounter{numexercice}
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\newcommand{\thetitle}{Feuille 7. Matrices}
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\rhead{\textbf{\thepage/\pageref{fin}}}
\lfoot{\small \textit{Lycée Joachim du Bellay}}
\cfoot{\small \textit{Mathématiques, prépa ECE1}}
\rfoot{\small \textit{http://allken-bernard.org/pierre/ece}}
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%%%     Exercice 310
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soient~:

\[A=\begin{pmatrix}
1 & 3 \\
-2 & 4 \\
\end{pmatrix}\]

\[B=\begin{pmatrix}
0 & 2 \\
1 & 3 \\
\end{pmatrix}\]

\[C=\begin{pmatrix}
-1 & -2 \\
1 & 3 \\
\end{pmatrix}\]

Calculer~:
\[A+(B-2C)-((A-B+3C)-((A-B)-(2B-C)))\]
\end{exercice}


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%%%     Exercice 311
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soient~:

\[A=\begin{pmatrix}
1 & -1 & 1 \\
2 & 1 & 3 \\
0 & 2 & 0 \\
\end{pmatrix}\]

\[B=\begin{pmatrix}
-1 & 0 & 2 \\
0 & 1 & 1 \\
-1 & 1 & 1 \\
\end{pmatrix}\]

\[C=\begin{pmatrix}
1 & 2 & 3 \\
3 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}\]

Calculer~:
\[-(-(-A-(B+4C))-(4A-7B+8(C-A+B))+3A)-5A+2B-C\]
\end{exercice}


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%%%     Exercice 312
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\begin{exercice}
\nopagebreak[5]Résoudre l'équation d'inconnue $X\in \mathcal M_2(\mathbf R)$~:
\[
\begin{pmatrix} 2 & 3 \\ 1 & 1 \end{pmatrix}
+X = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 313
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Résoudre l'équation d'inconnue $X\in \mathcal M_2(\mathbf R)$~:
\[
\begin{pmatrix} 1 & 4 \\ 3 & -1 \end{pmatrix}+3X
 = -\begin{pmatrix} 4 & 1 \\ 2 & 6 \end{pmatrix}-2X
\]
\end{exercice}


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%%%     Exercice 314
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Résoudre l'équation d'inconnue $X\in \mathcal M_{3,2}(\mathbf R)$~:
\[
5\left(X-
\begin{pmatrix} 1 & 2 \\ 0 & 1 \\ 2 & 3 \end{pmatrix}\right)
-\left(2X+
\begin{pmatrix} 0 & 3 \\ 3 & 7 \\ 0 & 4 \end{pmatrix}\right)
=
\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{pmatrix}
-2X
\]
\end{exercice}


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%%%     Exercice 315
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Résoudre l'équation d'inconnue $X\in \mathcal M_2(\mathbf R)$ et où $m$ est un paramètre réel~:
\[
\begin{pmatrix} 1+m & 2 \\ -1 & 5 \end{pmatrix}
+(m^2-1)X
=
\begin{pmatrix} 2 & 3m-1 \\ -2m+1 & m^2+4 \end{pmatrix}
\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 316
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Résoudre le système suivant, où les inconnues $X$ et $Y$ sont des éléments de $\mathcal M_2(\mathbf R)$~:
\[
\left\{\begin{array}{ccccc}
X & - & Y & = & \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix} \\
X & + & Y & = & \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}
\end{array}\right.
\]
\end{exercice}


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%%%     Exercice 317
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Résoudre le système suivant, où les inconnues $X$ et $Y$ sont des éléments de $\mathcal M_2(\mathbf R)$~:
\[
\left\{\begin{array}{ccccc}
2X & - & Y & = & \begin{pmatrix} 1 & 3 \\ 2 & -4 \end{pmatrix} \\
-X & + & 2Y & = & \begin{pmatrix} -1 & 0 \\ 0 & 3 \end{pmatrix}
\end{array}\right.
\]
\end{exercice}


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%%%     Exercice 318
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Résoudre le système suivant, où les inconnues $X$ et $Y$ sont des éléments de $\mathcal M_2(\mathbf R)$ et où $a$ est un paramètre réel~:
\[
\left\{\begin{array}{ccccc}
aX & + & (a+1)Y & = & \begin{pmatrix} 1 & 4 \\ -1 & 7 \end{pmatrix} \\
X & + & Y & = & \begin{pmatrix} 1 & 4 \\ 2 & -1 \end{pmatrix}
\end{array}\right.
\]
\end{exercice}


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%%%     Exercice 319
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Résoudre le système suivant, où les inconnues $X$, $Y$, $Z$ sont des éléments de $\mathcal M_3(\mathbf R)$ (on explicitera les solutions)~:
\[
\left\{\begin{array}{rcrcrcl}
X & - & Y &   &   &  =  & I_3 \\
2X & + & Y & - & Z & = & 0_3 \\
X  & + & 2Y & + & Z & = & I_3 \\
X & + & Y & - & 2Z & = & 3I_3
\end{array}\right.
\]
\end{exercice}


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%%%     Exercice 320
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre le système suivant, où les inconnues $X$, $Y$, $Z$ sont des éléments de $\mathcal M_3(\mathbf R)$ (on explicitera les solutions)~:
\[
\left\{\begin{array}{rcrcrcl}
2X & - & Y & +  & Z  &  =  & 0_3 \\
X & + & Y & - & 2Z & = & I_3 \\
-X  & + & Y & + & Z & = & 0_3 \\
2X & + & Y &  &  & = & I_3
\end{array}\right.
\]
\end{exercice}


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%%%     Exercice 321
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
2 & 1 & 0 \\
\end{pmatrix}
\begin{pmatrix}
1 \\
2 \\
3 \\
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 322
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
-2 & 3 & 0 & 1 \\
 1 & 1 & 2 & -1 \\
\end{pmatrix}
\begin{pmatrix}
-2 & 0 \\
-1 & 1 \\
1 & -2\\
-1 & -3
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 323
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
1 & 3 \\
2 & 4 \\
\end{pmatrix}
\begin{pmatrix}
-1& 3 \\
-2 & 4\\
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 324
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
1 & 0 & 3 \\
-1 & 1 & 0 \\
1 & 2 & 4 \\
\end{pmatrix}
\begin{pmatrix}
1 & 1 & -1 \\
1 & -1 & 1 \\
-1 & 1 & 1
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 325
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3 \\
\end{pmatrix}
\begin{pmatrix}
5 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & -8 \\
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 326
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
3 \\
5 \\
7 \\
\end{pmatrix}
\begin{pmatrix}
213 & 510 & 128 \\
\end{pmatrix}
\begin{pmatrix}
-3 \\
1 \\
1 \\
\end{pmatrix}
\begin{pmatrix}
-2 & 3 & -1 \\
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 327
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[
\begin{pmatrix}
-1 & -2 & 0 & 1 \\
-2 & 0 & 1 & 2 \\
1 & -1 & -1 & -1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 & -1 & 1 \\
0 & 1 & 1 & 0 \\
0 & -4 & 0& 0 \\
1 & 2 & -1 & 1 \\
\end{pmatrix}
\]
\end{exercice}


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%%%     Exercice 328
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Trouver toutes les matrices de la forme :
\[A=\begin{pmatrix} a&b\\ a&b\end{pmatrix}\]
dont le carré est la matrice nulle.
\end{exercice}


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%%%     Exercice 329
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Trouver toutes les matrices de la forme :
\[A=\begin{pmatrix} a&b\\ a&b\end{pmatrix}\]
dont le carré est la matrice unité.
\end{exercice}


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%%%     Exercice 330
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Trouver toutes les matrices de la forme :
\[A=\begin{pmatrix} a&b\\ a&b\end{pmatrix}\]
dont le carré est égal à~$A$.
\end{exercice}


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%%%     Exercice 331
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soient :
\[A=\begin{pmatrix} 0&1\\ 1&0\end{pmatrix}\qquad\textrm{et}\qquad B=\begin{pmatrix} 0&1\\ -1&0\end{pmatrix}\]
Calculer $A^2 B^2$ et $(AB)^2$.
\end{exercice}


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%%%     Exercice 332
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 1&1\\ 0&1\end{pmatrix}\]
Démontrer que~:
\[\forall n\in\mathbf N,\; A^n=\begin{pmatrix} 1&n\\ 0&1\end{pmatrix}\]
\end{exercice}


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%%%     Exercice 333
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 1&1\\ 0&2\end{pmatrix}\]
Démontrer que~:
\[\forall n\in\mathbf N,\; A^n=\begin{pmatrix} 1&2^n-1\\ 0&2^n\end{pmatrix}\]
\end{exercice}


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%%%     Exercice 334
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 1&0&0\\ 1&1&0 \\ 0 & 1 & 1\end{pmatrix}\]
Démontrer que~:
\[\forall n\in\mathbf N,\; A^n=\begin{pmatrix} 1& 0 & 0 \\ n &  1 & 0 \\ \dfrac{n(n-1)}{2} & n & 1 \end{pmatrix}\]
\end{exercice}


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%%%     Exercice 335
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}\]
\begin{enumerate}
\item Exprimer $A^2$ en fonction de $A$.
\item En déduire que $A^n=3^{n-1}A$ pour tout $n\in\mathbf N^*$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 336
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit :
\[B=\begin{pmatrix} 1&2\\ -3&6 \end{pmatrix}\]
\begin{enumerate}
\item Montrer que $(B-3I_2)(B-4I_2)=0_{2}$.
\item Exprimer $B^2$ en fonction de $I$ et $B$.
\item Définir par récurrence deux suites de réels $(a_n)_{n\in\mathbf N}$ et $(b_n)_{n\in\mathbf N}$ telles que :
\[\forall n\in\mathbf N,\quad B^n=a_n B +b_n I_2\]
\item Calculer $a_n$ et $b_n$ en fonction de $n$ puis expliciter $B^n$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 337
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\]
\begin{enumerate}
\item Calculer $A^4$.
\item Soit $n\in\mathbf N$. Soient $q$ et $r$ le quotient et le reste de la division euclidienne de $n$ par $4$. On a donc $n=4q+r$ et $r\in \{0,1,2,3\}$.
Exprimer $A^n$ en fonction de $A^r$.
\item En déduire $A^{51}$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 338
%%%     Le numéro correspond à la base de données :
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 1 & \frac{1}{2} \\ 2 & 1 \end{pmatrix}\]
\begin{enumerate}
\item Montrer qu'il existe deux suites $(a_n)$ et $(b_n)$ telles que, pour tout $n\in\mathbf N$~:
\[A^n=\begin{pmatrix} a_n & \frac{1}{2}b_n \\ 2b_n & a_n \end{pmatrix}\]
\item Calculer $a_n+b_n$ et $a_n-b_n$ en fonction de $n$.
\item En déduire $A^n$ en fonction de $n$.
\end{enumerate}
\end{exercice}


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%%%     Exercice 339
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit :
\[A=\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}\]
Calculer $A^n$ pour $n\in\mathbf N$.
\end{exercice}


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%%%     Exercice 340
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]La matrice suivante est-elle inversible~? Si oui calculer son inverse.
\[A=\left(\begin{matrix}
-2 & 3 & 1 \\
3 & 6 & 2\\
1 & 2 & 1
\end{matrix}\right)\]
\end{exercice}


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%%%     Exercice 341
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]La matrice suivante est-elle inversible~? Si oui calculer son inverse.
\[A=\left(\begin{matrix}
2 & 2 & -1\\
2 & -1 & 2\\
-1 & 2 & 2
\end{matrix}\right)\]
\end{exercice}


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%%%     Exercice 342
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]La matrice suivante est-elle inversible~? Si oui calculer son inverse.
\[A=\left(\begin{matrix}
1 & -3 & -1\\
-2 & 7 & 2\\
3 & 2 & 3
\end{matrix}\right)\]
\end{exercice}


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%%%     Exercice 343
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]La matrice suivante est-elle inversible~? Si oui calculer son inverse.
\[A=\left(\begin{matrix}
2 & 1 & -1\\
3 & 1 & -2\\
1 & 0 & 1
\end{matrix}\right)\]
\end{exercice}


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%%%     Exercice 344
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]La matrice suivante est-elle inversible~? Si oui calculer son inverse.
\[A=\left(\begin{matrix}
0 & 1 & 1 & 1\\
-1 & 0 & 1 & 1\\
-1 & -1 & 0 & 1\\
-1 & -1 & -1 & 0
\end{matrix}\right)\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 345
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]La matrice suivante est-elle inversible~? Si oui calculer son inverse.
\[A=\left(\begin{matrix}
1 & 1 & 1 & 0\\
-1 & 2 & 1 & 0\\
1 & 4 & 1 & 0\\
0&0&0&3
\end{matrix}\right)\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 346
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\begin{enumerate}
\item Montrer que la matrice $A=\left(\begin{matrix} 4& 11\\ 1& 3\end{matrix}\right)$ est inversible et calculer $A^{-1}$.
\item Résoudre l'équation d'inconnue $M\in\mathcal M_2(\mathbf R)$~:
\[\left(\begin{matrix} 4& 11\\ 1& 3\end{matrix}\right)M+\left(\begin{matrix} 1& -1\\ 1& 2\end{matrix}\right)=\left(\begin{matrix} 1& 1\\ -3& 5\end{matrix}\right)\]
\end{enumerate}
\end{exercice}


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%%%     Exercice 347
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soient $A,P\in\mathcal M_n(\mathbf R)$. On suppose que $P$ est inversible.
Démontrer que~
\[\forall n\in\mathbf N,\; (P^{-1}A P)^n=P^{-1}A^n P\]
On en donnera deux démonstrations~: l'une par récurrence, l'autre par un calcul \og{}direct\fg{}.
\end{exercice}


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\begin{exercice}
\nopagebreak[5]Dire pour chaque système s'il est de Cramer. Lorsque c'est le cas, le résoudre par une inversion de matrice.
\[\begin{array}{cc}
\left\{\begin{array}{ccc}
x+y&=&2\\
x-y&=&1
\end{array}\right.
&
\left\{\begin{array}{ccc}
2x-4y&=&5\\
-x+2y&=&3
\end{array}\right.\\\\
\left\{\begin{array}{ccc}
11 x + 10y&=&26\\
x+y&=&9\\
\end{array}\right.
&
\left\{\begin{array}{ccc}
13 x + 2y&=&0\\
6x+y&=&-4\\
\end{array}\right.
\end{array}\]
\end{exercice}


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\begin{exercice}
\nopagebreak[5]Soient $A\in\mathcal M_n(\mathbf R)$ et $r\in\mathbf N^*$.
Démontrer que si $A^r=0_n$, alors $I_n-A$ est inversible et son inverse est $I_n+A+A^2+\cdots+A^{r-1}$. Calculer ainsi l'inverse de la matrice :
\[\left(\begin{matrix}
1 & -1 & -1 \\
0 & 1 & -1 \\
0 & 0 & 1 \\
\end{matrix}\right)\]
\end{exercice}


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\begin{exercice}
\nopagebreak[5]Soit $A=\begin{pmatrix} 1&a&1\\ 1&2a&1\\ 1&1&b\end{pmatrix}$. Pour quelles valeurs de $a$ et $b$ la matrice $A$ est-elle inversible~? Donner alors son inverse.


\end{exercice}


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\begin{exercice}
\nopagebreak[5]\begin{enumerate}
\item Soit $A=\begin{pmatrix} 0&1&1\\ 1&0&1\\ 1&1&0\end{pmatrix}$. Démontrer que cette matrice est inversible et calculer son inverse. 
\item Résoudre le système linéaire :
\[(S)\left\{\begin{array}{cccc}
&y&+z&=1\\
x&&+z&=2\\
x&+y&&=3\\
\end{array}\right.\]
\end{enumerate}
\end{exercice}

\label{fin}
\end{document}