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Mathématiques - Chapitre 3 "Matrices"
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Ce fichier contient le 3ème Chapitre du cours Mathématiques.
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CHAPITRE IIIALGÈBRE DES MATRICES
I. DÉFINITION ET PREMIÈRES POPRIÉTÉ
Soient 𝒏, 𝒌 ∈ ℕ∗
On appelle matrice (réelle) de taille 𝒏 × 𝒌 tout tableau de nombres réels de 𝒏 lignes et 𝒌 colonnes.
On notera 𝑴𝒏𝒌 l’ensemble des matrices de taille 𝒏 × 𝒌.
Soit 𝑨 ∈ 𝑴𝒏𝒌 , on notera pour tout (𝒊, 𝒋) ∈ {𝟏, . . . , 𝒏} × {𝟏, . . . , 𝒌}, 𝒂𝒊𝒋 le coefficient de la 𝒊ème ligne
et 𝒋ème colonne de 𝑨.
𝑨 = (𝒂𝒊𝒋 )𝒊&𝟏…𝒏 ; 𝒋&𝟏…𝒌
EXEMLPES :
1
2 3
478 ∈ 𝑀+, : = ∈ 𝑀-- (−13 2 5 6) ∈ 𝑀,.
7 −1
3
1 3 7 𝑎-+ = −7
𝐴=: = ∈ 𝑀-+ 𝑎,- = 3
1 2 −7
A. Opération sur les matrices
Addition de deux matrices : Soient 𝑨, 𝑩 ∈ 𝑴𝒏𝒌 , on définit la matrice 𝑨 + 𝑩 de 𝑴𝒏𝒌 par :
(𝒂 + 𝒃)𝒊𝒋 = 𝒂𝒊𝒋 + 𝒃𝒊𝒋 𝒊 = 𝟏 … 𝒏, 𝒋 = 𝟏…𝒌
EXEMLPE :
1 2 2 7 3 9
: =+: ==: =
0 −1 −1 3 −1 2
1 0 1
428 + 418 = 428
3 1 4
Produit par un scalaire : Soient 𝑨 ∈ 𝑴𝒏𝒌 et 𝝀 ∈ ℝ, on définit la matrice 𝝀. 𝑨 de 𝑴𝒏𝒌 par :
(𝝀. 𝒂)𝒊𝒋 = 𝝀 × 𝒂𝒊𝒋 𝒊 = 𝟏 … 𝒏, 𝒋 = 𝟏…𝒌
EXEMLPE :
1 3 1 2 6 2
2×: = ∈ 𝑀-+ = : = ∈ 𝑀-+
0 1 −1 0 2 −2
1 −3
(−3) × 478 ∈ 𝑀+, = 4−218 ∈ 𝑀+,
6 −18
On a alors
PROPOSITION : (𝑴𝒏𝒌 , +, ∙) est un espace vectoriel.
Notation : On notera 𝟎𝒏𝒌 le vecteur nul de 𝑴𝒏𝒌 c’est-à-dire la matrice de taille 𝒏 × 𝒌 remplie de zéros.
EXEMPLE :
0 0 5 3 5 3
: =+: ==: =
0 0 2 1 2 1
, PROPOSITION : Pour 𝒊 = 𝟏 … 𝒏, 𝒋 = 𝟏 … 𝒌 soit 𝑬𝒊𝒋 la matrice dont tous les coefficients sont nuls sauf celui
de la 𝒊ème ligne et 𝒋ème colonne qui vaut 1.
Alors O𝑬𝒊𝒋 , 𝒊 = 𝟏. . . 𝒏, 𝒋 = 𝟏. . . 𝒌P = {𝑬𝟏𝟏 , … 𝑬𝟏𝒌 , 𝑬𝟐𝟏 , … … … 𝑬𝒏𝒌 } est une base de 𝑴𝒏𝒌 .
En particulier 𝒅𝒊𝒎 𝑴𝒏𝒌 = 𝒏 × 𝒌.
EXEMPLE :
1 0 0 1 0 0 0 0
ST U,T U,T U,T UV est une base de 𝑴𝟐𝟐 .
0 0 0 0 1 0 0 1
𝑎 𝑏 𝑎 𝑏 1 0 0 1 0 0 0 0
• ∀: = ∈ 𝑀-- : = = 𝑎: =+𝑏: =+𝑐: =+𝑑: =
𝑐 𝑑 𝑐 𝑑 0 0 0 0 1 0 0 1
1 0 0 1 0 0 0 0
𝑀-- = 𝑉𝑒𝑐𝑡 S: =,: =,: =,: =V Þ Famille génératrice
0 0 0 0 1 0 0 1
1 0 0 1 0 0 0 0 0 0
• Si 𝑎 : =+𝑏: =+𝑐: =+𝑑: ==
0 0 0 0 1 0 0 1 0 0
𝑎 𝑏 0 0
: ==: = Þ Famille génératrice
𝑐 𝑑 0 0
à𝑎 =𝑏 =𝑐 =𝑑 =0
⚠ : Le produit est plus dur à définir, on fait le produit de deux matrices de tailles différentes :
Produit de deux matrices : Soient 𝑨 ∈ 𝑴𝒏𝒌 et 𝑩 ∈ 𝑴𝒌𝒍 , on définit 𝑨𝑩 comme la matrice de 𝑴𝒏𝒍 donc le
coefficient (𝒊, 𝒋) est donné par :
𝒌
(𝒂 × 𝒃)𝒊𝒋 = ^ 𝒂𝒊𝒎 𝒃𝒊𝒋 + ⋯ + 𝒂𝒊𝒌 𝒃𝒌𝒋
𝒎&𝟏
EXEMPLE :
(𝐴) × (𝐵) = (𝐴 × 𝐵)
𝑴𝒏𝒌 𝑴𝒌𝒍 𝑴𝒏𝒍
EXEMPLE :
1 3 0 2=2×1+1×0
×: =
0 1 0 5=2×3+1×1
2 1 2 5 0
: ==: = 1=1×1+3×0
1 3 1 0 0
REMARQUE :
∀𝑨 ∈ 𝑴𝒏𝒌 , 𝑨 × 𝟎𝒌𝒍 = 𝟎𝒏𝒍
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