Chapitre 13 : Matrices
30 50 20
40 30 15
25 20 10
1 30 20
1 20 15
3 200 200
140 5900 5350
115 4800 4250
75 3150 2800
30 ×1 + 50 ×
1 + 20 ×3
n p n p N∗
n p A = (aij )16i6n
16j6p
A=
a11 a12 . . . a1n
a21 a2n
an1. . . . . . ann
aij A i
j
n p Mn,p(R)
n=p
n n Mn(R)
n= 1
p= 1
0n,p 0
n p
Mn(R)In=
1 0 . . . . . . 0
0 1 0 . . . 0
0
0. . . . . . 0 1
A B Mn,p(R)A B
A+B=M mi,j =ai,j +bi,j
•(A+B) + C=A+ (B+C)
•A+B=B+A
•0 + A=A+ 0 = A
•A B A +B=B+A= 0
−A A
A=
2 3 −1
063
−4 1 −2
B=
−3 0 0
5−2 7
4−1−1
A+B=
−1 3 −1
5 4 10
0 0 −3
A λ λA
A λ
λ(A+
B) = λA +λB ∀A∈ Mn(R),1.A =A∀(λ, µ)∈
R2, λ(µA) = (λµ)A
A=3−1
2 8 −2A=−6 2
−4−16
A∈ Mn,m(R)B∈ Mm,p(R)A B
A×B=M∈ Mn,p(R)∀i∈ {1, . . . , n} ∀j∈ {1, . . . , p}mij =
m
X
k=1
aikbkj
i A j
B
•(AB)C=A(BC)
•A(B+C) = AB +AC
(A+B)C=AC +BC
• ∀A∈ Mn,p(R)InA=AIp=A
•
InA=A I In
mij i, j IA mij =
n
X
k=1
IikAkj
Iik Iii 1mij =Aij
Ip
•AB
BA
AB 6=BA A B
•
•AB =AC
B6=C AB = 0 A6= 0 B6= 0
•
A=3 4 −1
2 0 5 B=
−2 6
1−1
2 3
A×B=−4 11
6 27
A=4−2
−2 1 B=1−3
2−6A×B= 0
A∈ Mn,p(R)M∈ Mp,n(R)mij =
aji tA A tA
∀A, B ∈ Mn,p(R)
•t(tA) = A
•t(A+B) = tA+tB
•t(λA) = λtA
• ∀C∈ Mp,m(R)t
(AC) = tCtA
ij
t(AC)ji AC
p
X
k=1
AjkCki
p
X
k=1
(tC)ik(tA)kj =
p
X
k=1
CkiAjk
A=
4 3 −2
051
−1 1 8
tA=
4 0 −1
351
−2 1 8
A∈ Mn(R)A=tA∀i, j ∈ {1, . . . , n}
aij =aji
aii
A A =
a11 0. . . 0
0a22
0
0. . . 0ann
∀i, j ∈ {1, . . . , n}, i > j ⇒aij = 0
A=
a11 a12 . . . a1n
0a22
0. . . 0ann
n A B
ij AB
n
X
k=1
aikbkj
k=i k =j
i6=j
AB
A B i > j ij AB
n
X
k=1
aikbkj =
i−1
X
k=1
0×bkj +
n
X
k=i
aik×0 =
0AB
A=
2 3 −1
0 6 3
0 0 −2
B=
1−5 2
0 3 −3
0 0 1
A×B=
2−1−6
0 18 −15
0 0 −2
A×B A
B
A∈ Mn(R)A A0=In
∀k>1, Ak=A×A× · · · × A
| {z }
k
Ak+m=AkAm(Ak)m=Akm n m λ
(λA)k=λkAk
(AB)k6=AkBkA B
A n An= 0
A B
Mn(R) (A+B)k=
k
X
i=0 k
iAiBk−i
A=
λ10. . . 0
0λ2
0
0. . . 0λn
Ak=
λk
10. . . 0
0λk
2
0
0. . . 0λk
n
B=
0 2 3
0 0 −1
0 0 0
B2=
0 0 −2
0 0 0
0 0 0
∀k>3, Bk= 0
C=
2 2 3
0 2 −1
0 0 2
C= 2I3+B I3B
Ak= (2I3+B)k= (2I3)k+k×(2I3)k−1B+k(k−1)
2×(2Ik)k−2B2= 2kI3+k2k−1B+
k(k−1)
22k−2B2C4= 16I3+ 32B+ 24B2=
16 64 48
0 16 −32
0 0 16
6
7
1
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7
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