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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}
