Name: Cours template by - Rezak Souilah
Author: MOSAID Articles & Tutorials

Cours template by - Rezak Souilah

📅 March 21, 2024 | 👁️ Views: 714 | 📝 2 exercises

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• ملف الدرس بصيغة PDF معروض أدناه — يمكنك تصفحه والاطلاع عليه مباشرة دون الحاجة لتحميله.

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\documentclass[12pt,a4paper]{book}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage[left=3cm,right=2.5cm,top=2.5cm,bottom=2.5cm]{geometry}
\usepackage[svgnames,x11names]{xcolor}
\usepackage{colortbl}
\usepackage{fancyhdr}
\usepackage[explicit]{titlesec}
\usepackage{tikz}
\usetikzlibrary{matrix,fit,calc,shadows,shadows.blur,shapes.symbols,positioning}
\usepackage[most]{tcolorbox}
\usepackage{pifont}
\usepackage{multicol}
\usepackage{enumerate}
\usepackage[shortlabels,inline]{enumitem}
\usepackage{eso-pic}
\usepackage{tikzpagenodes}
\usepackage{nopageno}
%===========================================================
%définition des couleurs
\definecolor{myblueii}{RGB}{63,200,244}
\definecolor{MainRed}{rgb}{.6, .1, .1}
\definecolor{mybluei}{RGB}{0,173,239}
\definecolor{bubblegum}{rgb}{0.99, 0.76, 0.8}
\definecolor{clight2}{RGB}{212, 237, 244}
\definecolor{blueslategray}{cmyk}{0.189, 0.091, 0, 0.439}
\definecolor{greenbrown}{HTML}{555544}
\definecolor{cottonseed}{cmyk}{0, 0.026, 0.047, 0.255}
\definecolor{lightblue}{cmyk}{0.272, 0.068, 0, 0.192}
\definecolor{bluetrout}{cmyk}{0.18, 0.15, 0, 0.608}
\definecolor{brownred}{cmyk}{0, 0.531, 0.557, 0.239}
\definecolor{orangewhite}{cmyk}{0, 0.031, 0.075, 0.004}
\definecolor{avocado}{cmyk}{0.06, 0, 0.55, 0.37}
\definecolor{eggplant}{cmyk}{0.04, 0.24, 0, 0.47}
\definecolor{darkblue}{rgb}{0.12,0.47,0.87}
%===============================================================
%===========================================
\definecolor{shadow@color}{cmyk}{.07,0,0,0.49}
\definecolor{col1}{RGB}{12, 102, 98}
\definecolor{col2}{RGB}{248, 193, 12}
\definecolor{col3}{RGB}{236, 67, 1}
%========================================
%=================================================
\definecolor{gray}{RGB}{236,236,236}
\definecolor{shadow@color}{cmyk}{.07,0,0,0.49}
\definecolor{section@title@color}{cmyk}{1,0.2,0.3,0.1}
\definecolor{subsection@title@color}{cmyk}{0,0.6,0.9,0}
%=================================================
%========================================
%Commands
%newcommand % % % % %
\newcommand\z{\cellcolor{bubblegum}}
\newcommand\y{\cellcolor{clight2}}
% % % % % % % % % % % % % % % % % % % % % % %
\newcommand\highlight[1]{\tikz[overlay, remember picture,baseline=-\the\dimexpr\fontdimen5\textfont2\relax]\node[rectangle,fill=blue!50,rounded corners,fill opacity = 0.2,draw,thick,text opacity =1] {$#1$};}
% % % % % % % % % % % % % % % % % % % % % % %
\newcommand{\tikzmark}[2]{\tikz[overlay,remember picture,baseline=(#1.base)] \node (#1) {#2};}
% % % % % % % % % % % % % % % % % % % % % % %
\newcommand\tikznode[3][]%
{\tikz[remember picture,baseline=(#2.base)]
 \node[minimum size=0pt,inner sep=0pt,#1](#2){#3};%
}
%==============================================
%=============================================
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%===========================================================
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%============================================
%==============================================
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{\begin{tikzpicture}[remember picture, overlay]%
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fill=section@title@color,
anchor=north west,
text width=\paperwidth,
text height=2cm,
text depth=2cm,
inner xsep=1cm,
font=\color{white}\bigsf
]
at ([yshift=-2.5cm]current page.north west) (blackrect) {#1};
% the khaki stripe
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% the banner
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signal,
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% the text in the banner
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at ([yshift=1.5cm]banner.east) {\chaptertitlename};
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{1ex}
{}
[]
%------------------------------------------------------
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%================================================
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%======================================================
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\titleformat{\subsection}
{\subsectiontitle@font\color{subsection@title@color}}{}{0em}
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%===============================================================
%=============================================
% ===== Environment "definition" ======================
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%=====================================================
% % % % % %% Example environment
\newcounter{examp}
\setcounter{examp}{0}
\newcommand{\mytitle}[2]%
{%
 \stepcounter{examp}
 \begin{tikzpicture}[font=\sffamily]
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 \end{tikzpicture}
}
%==============================================================
%Exercice résolu
\makeatletter
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 sharp corners,
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 attach boxed title to top left,
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 empty,
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 \draw[orange!80!black, line width=.5mm] (title.south)|-(frame.north east);
 },
 #1
}
\makeatother
%===============================================================
%Exercice
%commands
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}
\setlength{\columnsep}{30pt}
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%============================================================
%======================================================
%list
%========================================
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%=======================================
\newlist{listexos}{enumerate}{2} % en espérant qu’il n’ait pas plus d’un niveau de sous-questions
\setlist[listexos,1]{label =\lblbox{\arabic*}, font=\itshape\bfseries\large, wide=0pt, leftmargin=*}
%============================================================
%============================================================
%================================================================
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%===============================================================
%============================================================
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%======================================================
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%==============
%======================================================
\begin{document}
\chapter{Matrices}
\section{Notions générales}
\begin{definition}
 Une Matrice est un tableau rectangulaire de la forme
 \begin{equation*}
 A =
 \left[
 \begin{array}{ccc>{\columncolor{clight2}}c cc}
 a_{11} & a_{12} & \cdots &  a_{1j} & \cdots & a_{1n}\\
 a_{21} & a_{22} & \cdots &  a_{2j} & \cdots & a_{2n}\\
 \vdots & \vdots &\vdots & \vdots & \vdots&\vdots \\
 \rowcolor{clight2} a_{i1} & a_{i2} & \cdots & \cellcolor[gray]{.6}a_{ij} & \cdots & a_{in}\\
 \vdots & \vdots &\ddots & \vdots & \vdots &\vdots \\
 a_{m1} & a_{m2} & \cdots & a_{mj} & \cdots & a_{mn}
 \end{array}
 \right]
 \end{equation*}
 où les $a_{ij}$ sont des nombres réels appelés les éléments ou coefficients de la matrice $A$. La matrice précédente est aussi notée par $\left(a_{ij}\right),\;i=1,\cdots,m,\;j=1,\cdots,n$, ou simplement par $\left(a_{ij}\right)$. L'élément $a_{ij}$ est situé à l’intersection de la $i$-ème ligne et de la $j$-ème colonne. Une matrice ayant $m$ lignes et $n$ colonnes
 est appelée une matrice d'ordre $(m,n)$, ou de dimension $m\times n$. Les matrices seront notées habituellement par des lettres capitales $A,\,B,\cdots$, et les éléments par des lettres minuscules $a,\,b,\cdots$.
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
On considère les matrices suivantes:
\begin{equation*}
A=
\begin{bmatrix}
1 & -1 & 2\\
3 &  \frac{1}{2} & \sqrt{2}
\end{bmatrix},\;
B=
\begin{bmatrix}
-\sqrt{3} \\
\frac{1}{5}\\
1
\end{bmatrix},\;
C=
\begin{bmatrix}
-1 & 0 & 1 & 5
\end{bmatrix},\;
D=
\begin{bmatrix}
-1 & 1 & 3 \\
5 & 6 & 2 \\
1 & -1 & 0
\end{bmatrix}.
\end{equation*}
\begin{listexos}
 \item La matrice $A$ est de dimension $2\times 3$ et on a $a_{23}=\sqrt{2},\quad a_{13}=2,\quad a_{22}=\frac{1}{2}$.
 \item La matrice $B$ est de dimension $3\times 1$ et on a $b_{11}=-\sqrt{3},\quad b_{21}=\frac{1}{5},\quad b_{31}=1$.
 \item La matrice $C$ est de dimension $1\times 4$ et on a $C_{11}=-1,\quad C_{12}=0,\quad C_{13}=1,\quad C_{14}=5$.
 \item La matrice $D$ est de dimension $3\times 3$ et on a $D_{33}=0,\quad D_{23}=2,\quad D_{32}=-1$.
\end{listexos}
\subsection{Égalité de deux matrices}
\begin{definition}
 Deux matrices $A$ et $B$ sont égales, et on écrit
 $A = B$, si elles ont même dimension et si leurs éléments correspondants sont égaux.
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
On considère les matrices $
A=
\begin{bmatrix}
0 & 1  \\
2 & 8
\end{bmatrix},
\;
B=
\begin{bmatrix}
0 & x^{2}\\
\sqrt{y} & z
\end{bmatrix}$.
\begin{itemize}
 \item Trouver $x,\,y\in\mathbb{R}$ tels que $A=B$.
\end{itemize}
\vspace{1cm}
\begin{mybox}{Exercice résolu}
 Trouver les valeurs possibles de $x,\,y\in\mathbb{R}$ telles que les matrices
 \[
 E=
 \begin{bmatrix}
 x^{2}-5 & 1 \\
 3 & -2y-4
 \end{bmatrix}
 \quad
 F=
 \begin{bmatrix}
 1 & 1 \\
 x+2 & 0
 \end{bmatrix}.
 \]
 soit égales.
\end{mybox}
\section{Matrices particulières}
Dans cette on va définir quelques matrices spéciales.
\subsection{Matrice ligne}
\begin{definition}
 Une matrice ligne est une matrice comportant une seule ligne. Une matrice ligne a donc
 pour dimension $1\times n$. Une matrice ligne a la forme suivante:
 \begin{equation*}
 A=
 \begin{bmatrix}
 a_{11} & a_{12} & \cdots & a_{1n}
 \end{bmatrix}_{1\times n}.
 \end{equation*}
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
La matrice
$
A=
\begin{bmatrix}
1 & 2 & -1 & 5 & 0
\end{bmatrix}
$ est une matrice ligne de dimension $1\times 5$.
\subsection{Matrice colone}
\begin{definition}
 Une matrice colonne est une matrice comportant une seule colonne.  Une matrice colonne a donc
 pour dimension $m\times 1$. Une matrice colonne a la forme suivante:
 \begin{equation*}
 A=\begin{bmatrix}
 a_{11}  \\
 a_{21}  \\
 \vdots  \\
 a_{n1}
 \end{bmatrix}_{n\times 1}.
 \end{equation*}
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
La matrice
$
A=
\begin{bmatrix}
-1 \\
0 \\
2 \\
4
\end{bmatrix}$ est une matrice colonne de dimension $4\times 1$.
\section{Opérations sur les matrice}
Dans cette section on va définir les opérations algébriques sur les matrices.
\subsection{Somme de deux matrices}
\begin{definition}
 Soit $A$ et $B$ deux matrices de même dimension. La somme de $A$ et $B$, écrite $A+B$, est la matrice obtenue en ajoutant les éléments correspondants des deux matrices.
 \begin{equation*}
 \begin{aligned}
 &\text{Si}\quad A=\begin{bmatrix}
 a_{11} & a_{12} & a_{13}\\
 a_{21} & a_{22}  & a_{23}
 \end{bmatrix}_{2\times 3}\;\text{et}\quad
 B=\begin{bmatrix}
 b_{11} & b_{12} & b_{13}\\
 b_{21} & b_{22}  & b_{23}
 \end{bmatrix}_{2\times 3},
 \\ \ \\
 &\text{alors}\quad A+B=\begin{bmatrix}
 a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13}\\
 a_{21}+b_{21} & a_{22}+b_{22}  & a_{23}+b_{23}
 \end{bmatrix}_{2\times 3}.
 \end{aligned}
 \end{equation*}
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
On considère les matrices:
\begin{equation*}
A=
\begin{bmatrix}
-2 & 0 & 1\\
1 & 3  & -1\\
\end{bmatrix},\;
B=
\begin{bmatrix}
1 & 2 \\
1 & 0 \\
0 & 4
\end{bmatrix},\;
C=
\begin{bmatrix}
-2 & 1 \\
4 & 3 \\
-1 & -1
\end{bmatrix},\;
D=
\begin{bmatrix}
1 & 1 & 0\\
2 & 0 & 4
\end{bmatrix}.
\end{equation*}
\begin{listexos}
 \item Calchler $B+C$.
 \begin{equation*}
 B+C=\begin{bmatrix}
 1 & 2 \\
 1 & 0 \\
 0 & 4
 \end{bmatrix}+
 \begin{bmatrix}
 -2 & 1 \\
 4 & 3 \\
 -1 & -1
 \end{bmatrix}=
 \begin{bmatrix}
 1+(-2) & 2+1 \\
 1+4 & 0+3 \\
 0+1 & 4+(-1)
 \end{bmatrix}=
 \begin{bmatrix}
 -1 & 3 \\
 5 & 3 \\
 1 & 3
 \end{bmatrix}.
 \end{equation*}
 \item Calculer $A+D$.
 \begin{equation*}
 A+D=\begin{bmatrix}
 -2 & 0 & 1 \\
 1 & 3  & -1
 \end{bmatrix}+
 \begin{bmatrix}
 1 & 1 & 0\\
 2 & 0 & 4
 \end{bmatrix}=
 \begin{bmatrix}
 -2+1 & 0+1 & 1+0\\
 1+2 & 3+0 & -1+4
 \end{bmatrix}=
 \begin{bmatrix}
 -1 & 1 & 1\\
 3 & 3 & 3
 \end{bmatrix}.
 \end{equation*}
 \item  La somme de $A$ et $B$ n’est pas définie car $A$ et $B$ ne sont pas de même dimension.
\end{listexos}
\newpage
\section{Exercices}
\begin{multicols*}{2}
 \sect{Calcul matriciel}
 \begin{Exercice}
 On considère les matrices
 \[
 A=\begin{pmatrix}
 4 & 8 \\
 1 & 2
 \end{pmatrix},\;
 B=\begin{pmatrix}
 3 & 9 \\
 1 & 1
 \end{pmatrix}.
 \]
 \begin{listexos}
 \item Calculer $A+B,\,AB,\;BA,\,A^2$ et $B^2$.
 \item A-t-on $(A+B)^2=A^2+2AB+B^2$?
 \item Mêmes questions pour les matrices
 \[
 A=\begin{pmatrix}
 1 & 0 \\
 2 & 1
 \end{pmatrix},\;B=\begin{pmatrix}
 2 & 0 \\
 1 & 2
 \end{pmatrix}.
 \]
 \end{listexos}
 \end{Exercice}
 \begin{Exercice}
 On considère les matrices
 \[
 A=\begin{pmatrix}
 x & 5 \\
 0 & 2x
 \end{pmatrix},\;B=\begin{pmatrix}
 y & 7 \\
 -1 & 3y
 \end{pmatrix}.
 \]
 \begin{listexos}
 \item  Trouver $x,y\in\mathbb{R}$ tels que
 \[
 A+B=\begin{pmatrix}
 4 & 12 \\
 -1 & 17
 \end{pmatrix}.
 \]
 \item Trouver $x,y\in\mathbb{R}$ tels que
 \[
 2A-4B=\begin{pmatrix}
 -5 & -18 \\
 4 & -16
 \end{pmatrix}.
 \]
 \end{listexos}
 \end{Exercice}
 \begin{Exercice}
 On considère la matrice
 \[
 A=
 \begin{pmatrix}
 x & 1 \\
 2 & 3
 \end{pmatrix},\quad x\in\mathbb{R}.
 \]
 \begin{itemize}
 \item Trouver $x$ tel que $A^2=\begin{pmatrix}
 6 & 1 \\
 2 & 11
 \end{pmatrix}$
 \end{itemize}
 \end{Exercice}
 \begin{Exercice}
 On considère les matrices
 \[
 \begin{aligned}
 &A=
 \begin{pmatrix}
 1 & 3 \\
 -4 & 2 \\
 0 & 7
 \end{pmatrix},
 \;
 B=
 \begin{pmatrix}
 -2 & 0 \\
 -2 & 1 \\
 8 & 1
 \end{pmatrix},
 \\
 &C=
 \begin{pmatrix}
 -4 & 6 \\
 -14 & 7 \\
 24 & 17
 \end{pmatrix}.
 \end{aligned}
 \]
 \begin{itemize}
 \item Trouver $x, y\in\mathbb{R}$ tels que $xA+yB=C$.
 \end{itemize}
 \end{Exercice}
 \begin{Exercice}
 Calculer si possible les produits matriciels suivants:
 \begin{listexos}
 \item
 \[
 \begin{pmatrix}
 2 & 5 \\
 3 & 6\\
 4 & 7
 \end{pmatrix}
 \times
 \begin{pmatrix}
 2 & 5 \\
 4 & 6
 \end{pmatrix}
 \]
 \item
 \[
 \begin{pmatrix}
 -1 & 4 & 5
 \end{pmatrix}
 \times
 \begin{pmatrix}
 0 & 1 & 6\\
 3 & -1 & 4\\
 3 & 5 & -2
 \end{pmatrix}
 \]
 \item
 \[
 \begin{pmatrix}
 2 & -3 & 4\\
 -1 & 2& 6\\
 4 & -3 & -3
 \end{pmatrix}
 \times
 \begin{pmatrix}
 1 & 2 & -3\\
 0 & -4 & 1
 \end{pmatrix}
 ^T
 \]
 \end{listexos}
 \end{Exercice}
\end{multicols*}
\end{document}

    
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\begin{document}
\chapter{Matrices}
\section{Notions générales}
\begin{definition}
 Une Matrice est un tableau rectangulaire de la forme
 \begin{equation*}
 A =
 \left[
 \begin{array}{ccc>{\columncolor{clight2}}c cc}
 a_{11} & a_{12} & \cdots &  a_{1j} & \cdots & a_{1n}\\
 a_{21} & a_{22} & \cdots &  a_{2j} & \cdots & a_{2n}\\
 \vdots & \vdots &\vdots & \vdots & \vdots&\vdots \\
 \rowcolor{clight2} a_{i1} & a_{i2} & \cdots & \cellcolor[gray]{.6}a_{ij} & \cdots & a_{in}\\
 \vdots & \vdots &\ddots & \vdots & \vdots &\vdots \\
 a_{m1} & a_{m2} & \cdots & a_{mj} & \cdots & a_{mn}
 \end{array}
 \right]
 \end{equation*}
 où les $a_{ij}$ sont des nombres réels appelés les éléments ou coefficients de la matrice $A$. La matrice précédente est aussi notée par $\left(a_{ij}\right),\;i=1,\cdots,m,\;j=1,\cdots,n$, ou simplement par $\left(a_{ij}\right)$. L'élément $a_{ij}$ est situé à l’intersection de la $i$-ème ligne et de la $j$-ème colonne. Une matrice ayant $m$ lignes et $n$ colonnes
 est appelée une matrice d'ordre $(m,n)$, ou de dimension $m\times n$. Les matrices seront notées habituellement par des lettres capitales $A,\,B,\cdots$, et les éléments par des lettres minuscules $a,\,b,\cdots$.
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
On considère les matrices suivantes:
\begin{equation*}
A=
\begin{bmatrix}
1 & -1 & 2\\
3 &  \frac{1}{2} & \sqrt{2}
\end{bmatrix},\;
B=
\begin{bmatrix}
-\sqrt{3} \\
\frac{1}{5}\\
1
\end{bmatrix},\;
C=
\begin{bmatrix}
-1 & 0 & 1 & 5
\end{bmatrix},\;
D=
\begin{bmatrix}
-1 & 1 & 3 \\
5 & 6 & 2 \\
1 & -1 & 0
\end{bmatrix}.
\end{equation*}
\begin{listexos}
 \item La matrice $A$ est de dimension $2\times 3$ et on a $a_{23}=\sqrt{2},\quad a_{13}=2,\quad a_{22}=\frac{1}{2}$.
 \item La matrice $B$ est de dimension $3\times 1$ et on a $b_{11}=-\sqrt{3},\quad b_{21}=\frac{1}{5},\quad b_{31}=1$.
 \item La matrice $C$ est de dimension $1\times 4$ et on a $C_{11}=-1,\quad C_{12}=0,\quad C_{13}=1,\quad C_{14}=5$.
 \item La matrice $D$ est de dimension $3\times 3$ et on a $D_{33}=0,\quad D_{23}=2,\quad D_{32}=-1$.
\end{listexos}
\subsection{Égalité de deux matrices}
\begin{definition}
 Deux matrices $A$ et $B$ sont égales, et on écrit
 $A = B$, si elles ont même dimension et si leurs éléments correspondants sont égaux.
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
On considère les matrices $
A=
\begin{bmatrix}
0 & 1  \\
2 & 8
\end{bmatrix},
\;
B=
\begin{bmatrix}
0 & x^{2}\\
\sqrt{y} & z
\end{bmatrix}$.
\begin{itemize}
 \item Trouver $x,\,y\in\mathbb{R}$ tels que $A=B$.
\end{itemize}
\vspace{1cm}
\begin{mybox}{Exercice résolu}
 Trouver les valeurs possibles de $x,\,y\in\mathbb{R}$ telles que les matrices
 \[
 E=
 \begin{bmatrix}
 x^{2}-5 & 1 \\
 3 & -2y-4
 \end{bmatrix}
 \quad
 F=
 \begin{bmatrix}
 1 & 1 \\
 x+2 & 0
 \end{bmatrix}.
 \]
 soit égales.
\end{mybox}
\section{Matrices particulières}
Dans cette on va définir quelques matrices spéciales.
\subsection{Matrice ligne}
\begin{definition}
 Une matrice ligne est une matrice comportant une seule ligne. Une matrice ligne a donc
 pour dimension $1\times n$. Une matrice ligne a la forme suivante:
 \begin{equation*}
 A=
 \begin{bmatrix}
 a_{11} & a_{12} & \cdots & a_{1n}
 \end{bmatrix}_{1\times n}.
 \end{equation*}
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
La matrice
$
A=
\begin{bmatrix}
1 & 2 & -1 & 5 & 0
\end{bmatrix}
$ est une matrice ligne de dimension $1\times 5$.
\subsection{Matrice colone}
\begin{definition}
 Une matrice colonne est une matrice comportant une seule colonne.  Une matrice colonne a donc
 pour dimension $m\times 1$. Une matrice colonne a la forme suivante:
 \begin{equation*}
 A=\begin{bmatrix}
 a_{11}  \\
 a_{21}  \\
 \vdots  \\
 a_{n1}
 \end{bmatrix}_{n\times 1}.
 \end{equation*}
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
La matrice
$
A=
\begin{bmatrix}
-1 \\
0 \\
2 \\
4
\end{bmatrix}$ est une matrice colonne de dimension $4\times 1$.
\section{Opérations sur les matrice}
Dans cette section on va définir les opérations algébriques sur les matrices.
\subsection{Somme de deux matrices}
\begin{definition}
 Soit $A$ et $B$ deux matrices de même dimension. La somme de $A$ et $B$, écrite $A+B$, est la matrice obtenue en ajoutant les éléments correspondants des deux matrices.
 \begin{equation*}
 \begin{aligned}
 &\text{Si}\quad A=\begin{bmatrix}
 a_{11} & a_{12} & a_{13}\\
 a_{21} & a_{22}  & a_{23}
 \end{bmatrix}_{2\times 3}\;\text{et}\quad
 B=\begin{bmatrix}
 b_{11} & b_{12} & b_{13}\\
 b_{21} & b_{22}  & b_{23}
 \end{bmatrix}_{2\times 3},
 \\ \ \\
 &\text{alors}\quad A+B=\begin{bmatrix}
 a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13}\\
 a_{21}+b_{21} & a_{22}+b_{22}  & a_{23}+b_{23}
 \end{bmatrix}_{2\times 3}.
 \end{aligned}
 \end{equation*}
\end{definition}
\mytitle{SKILLS}{PROBLEM-SOLVING}
On considère les matrices:
\begin{equation*}
A=
\begin{bmatrix}
-2 & 0 & 1\\
1 & 3  & -1\\
\end{bmatrix},\;
B=
\begin{bmatrix}
1 & 2 \\
1 & 0 \\
0 & 4
\end{bmatrix},\;
C=
\begin{bmatrix}
-2 & 1 \\
4 & 3 \\
-1 & -1
\end{bmatrix},\;
D=
\begin{bmatrix}
1 & 1 & 0\\
2 & 0 & 4
\end{bmatrix}.
\end{equation*}
\begin{listexos}
 \item Calchler $B+C$.
 \begin{equation*}
 B+C=\begin{bmatrix}
 1 & 2 \\
 1 & 0 \\
 0 & 4
 \end{bmatrix}+
 \begin{bmatrix}
 -2 & 1 \\
 4 & 3 \\
 -1 & -1
 \end{bmatrix}=
 \begin{bmatrix}
 1+(-2) & 2+1 \\
 1+4 & 0+3 \\
 0+1 & 4+(-1)
 \end{bmatrix}=
 \begin{bmatrix}
 -1 & 3 \\
 5 & 3 \\
 1 & 3
 \end{bmatrix}.
 \end{equation*}
 \item Calculer $A+D$.
 \begin{equation*}
 A+D=\begin{bmatrix}
 -2 & 0 & 1 \\
 1 & 3  & -1
 \end{bmatrix}+
 \begin{bmatrix}
 1 & 1 & 0\\
 2 & 0 & 4
 \end{bmatrix}=
 \begin{bmatrix}
 -2+1 & 0+1 & 1+0\\
 1+2 & 3+0 & -1+4
 \end{bmatrix}=
 \begin{bmatrix}
 -1 & 1 & 1\\
 3 & 3 & 3
 \end{bmatrix}.
 \end{equation*}
 \item  La somme de $A$ et $B$ n’est pas définie car $A$ et $B$ ne sont pas de même dimension.
\end{listexos}
\newpage
\section{Exercices}
\begin{multicols*}{2}
 \sect{Calcul matriciel}
 \begin{Exercice}
 On considère les matrices
 \[
 A=\begin{pmatrix}
 4 & 8 \\
 1 & 2
 \end{pmatrix},\;
 B=\begin{pmatrix}
 3 & 9 \\
 1 & 1
 \end{pmatrix}.
 \]
 \begin{listexos}
 \item Calculer $A+B,\,AB,\;BA,\,A^2$ et $B^2$.
 \item A-t-on $(A+B)^2=A^2+2AB+B^2$?
 \item Mêmes questions pour les matrices
 \[
 A=\begin{pmatrix}
 1 & 0 \\
 2 & 1
 \end{pmatrix},\;B=\begin{pmatrix}
 2 & 0 \\
 1 & 2
 \end{pmatrix}.
 \]
 \end{listexos}
 \end{Exercice}
 \begin{Exercice}
 On considère les matrices
 \[
 A=\begin{pmatrix}
 x & 5 \\
 0 & 2x
 \end{pmatrix},\;B=\begin{pmatrix}
 y & 7 \\
 -1 & 3y
 \end{pmatrix}.
 \]
 \begin{listexos}
 \item  Trouver $x,y\in\mathbb{R}$ tels que
 \[
 A+B=\begin{pmatrix}
 4 & 12 \\
 -1 & 17
 \end{pmatrix}.
 \]
 \item Trouver $x,y\in\mathbb{R}$ tels que
 \[
 2A-4B=\begin{pmatrix}
 -5 & -18 \\
 4 & -16
 \end{pmatrix}.
 \]
 \end{listexos}
 \end{Exercice}
 \begin{Exercice}
 On considère la matrice
 \[
 A=
 \begin{pmatrix}
 x & 1 \\
 2 & 3
 \end{pmatrix},\quad x\in\mathbb{R}.
 \]
 \begin{itemize}
 \item Trouver $x$ tel que $A^2=\begin{pmatrix}
 6 & 1 \\
 2 & 11
 \end{pmatrix}$
 \end{itemize}
 \end{Exercice}
 \begin{Exercice}
 On considère les matrices
 \[
 \begin{aligned}
 &A=
 \begin{pmatrix}
 1 & 3 \\
 -4 & 2 \\
 0 & 7
 \end{pmatrix},
 \;
 B=
 \begin{pmatrix}
 -2 & 0 \\
 -2 & 1 \\
 8 & 1
 \end{pmatrix},
 \\
 &C=
 \begin{pmatrix}
 -4 & 6 \\
 -14 & 7 \\
 24 & 17
 \end{pmatrix}.
 \end{aligned}
 \]
 \begin{itemize}
 \item Trouver $x, y\in\mathbb{R}$ tels que $xA+yB=C$.
 \end{itemize}
 \end{Exercice}
 \begin{Exercice}
 Calculer si possible les produits matriciels suivants:
 \begin{listexos}
 \item
 \[
 \begin{pmatrix}
 2 & 5 \\
 3 & 6\\
 4 & 7
 \end{pmatrix}
 \times
 \begin{pmatrix}
 2 & 5 \\
 4 & 6
 \end{pmatrix}
 \]
 \item
 \[
 \begin{pmatrix}
 -1 & 4 & 5
 \end{pmatrix}
 \times
 \begin{pmatrix}
 0 & 1 & 6\\
 3 & -1 & 4\\
 3 & 5 & -2
 \end{pmatrix}
 \]
 \item
 \[
 \begin{pmatrix}
 2 & -3 & 4\\
 -1 & 2& 6\\
 4 & -3 & -3
 \end{pmatrix}
 \times
 \begin{pmatrix}
 1 & 2 & -3\\
 0 & -4 & 1
 \end{pmatrix}
 ^T
 \]
 \end{listexos}
 \end{Exercice}
\end{multicols*}
\end{document}

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