Fiche 5

M(p, n, )
(p, n)
A∈ M(p, n, )ai,j i`eme j`eme A= (ai,j )
M(p, n, )p n
p n ∆i,j
(i, j)∈ {1,· · · , p}×{1,· · · , n}∆i,j = (i, j) 1
p n
(i, j)p
A= (ai,j )
A=
p
X
i=1
n
X
j=1
ai,j ∆i,j 
A∈ M(p, n, )
A
V(L1(A),· · · , Lp(A)) M(1, n, )p A
A∈ M(p, n, )t A∈ M(n, p, )
A A = (ai,j )t A= (aj,i)
t(t
A) = A
t(αA +βB) = αt
A+βt B
(t A) = (A)
AV(C1(A),· · · , Cn(A))
M(p, 1,)n A
t
A=A
A= (ai,j ) (n, n)⇐⇒ ∀i, j ∈ {1,· · · , n}, aj,i =ai,j
(n, n)S(n, )n2+n
2M(n, )
t
A=−A
A= (ai,j ) (n, n)⇐⇒ ∀i, j ∈ {1,· · · , n}, aj,i =−ai,j
(n, n)A(n, )n2−n
2M(n, )
S(n, )A(n, )M(n, )
A∈ M(n, )A=1
2A+t
A+1
2A−t
A

k s1,· · · , sk∈r
t1,· · · , tr∈M= (mi,j )∈ M(k, r, )
s1
sk
=
m1,1t1+· · · +m1,rtr
mk,1t1+· · · +mk,r tr
.
s1
sk
=
m1,1· · · m1,r
mk,1· · · mk,r
t1
tr
s1
sk
=M
t1
tr
.
p z1,· · · , zp∈q y1,· · · , yq
n x1,· · · , xnz1,· · · , zp
x1,· · · , xnA∈ M(p, q, )B∈ M(q, n, )C∈ M(p, n, )
z1
zp
=A
y1
yq
y1
yq
=B
x1
xn
z1
zp
=C
x1
yn
z1
zp
=C
x1
xn
=A
y1
yq
=A B
x1
xn
C A B AB
A1· · · An
A1· · · AnA1
An
×(α1· · · αq)
β1
βq
= (α1β1+· · · +αqβq)
ci,j AB ×
ci,j =Li(A)×Cj(B)
∀A, B ∈ M(p, q, )∀M, N, ∈ M(q, n, )A(M+N) = AM +AN (A+B)M=AM +BM
AB A B
M(p, q, )p=q
InM(n, )
t(AB) = t
Bt A

Li(M)Cj(M)M
AB Li(AB) = Li(A)×B
AB B
AB Cj(AB) = A×Cj(B)
AB A
(AB)6min ( (A),(B))
Ik
N
M(p, q, )∀A∈ M(p, q, )N A =A
N(p, p)
A∆i,j M(p, q, )
N Ipp1
M(p, q, )Iq
A∈ M(p, q, )
A∈ M(p, q, )A0A0A=I
q q 6p
A∈ M(p, q, )A00 AA00 =I
p p 6q
A p
A−1
AA−1=A−1A=Ip
Ip
A
n
X
k=0
αkAk= 0 α06= 0
A A
A−1=−
n
X
k=1
αk
α0Ak−1

A∈ M(p, )
3A4−A3+ 5A2−7A1+ 2A0= 0
3A4−A3+ 5A2−7A1=−2Ip
−3
2A4+1 2A3−5 2A2+7 2A1=Ip
A
−3
2A3+1 2A2−5 2A1+7 2IpA=IpI1
A A−1=−3
2A3+1 2A2−5 2A+7 2Ip
A
A p
A AX =Y
X=A−1Y A−1
A p p
A A−1
(A|I)
AX =Y
(A|I)⇐⇒ · · · ⇐⇒ (I|A−1)
A A−1
A
A
(A|I)⇐⇒ (E1A|E1I)⇐⇒ · · · ⇐⇒ (Ek· · · E1A|Ek· · · E1I)=(I|A−1)
A−1=Ek· · · E1
A1· · · An
(A1· · · An)−1=A−1
n· · · A−1
1
A
Ek· · · E1A−1
A= (Ek· · · E1)−1=E−1
1· · · E−1
kE−1
jEj
A
A
A p M ∈ M(p, n, )
N∈ M(n, p, ) (AM) = (M) (N A) = (N)
A AM
NA
A
At At A−1=tA−1.
2 3
M(p, ) ( p)p
p
p
(p)pp−(p)p

1p(e1,· · · , ep)pp
A∈ M(p, )
det(A)|A|p A
p A
det(Ip)=1
A
A Li←λLiλ
A Li←Li+µLkk6=i
Adet(A) = 0
Adet(A)=0
Adet(A)
Adet(A)
C1det(t A) = det(A)
C2C62 6
p×p p (p−1) ×(p−1)
A= (ai,j )∈ M(p, )
Ac
i,j ∈ M(p−1,)i j A
ai,j (−1)i+jdet Ac
i,j 
i`eme
det(A) =
p
X
j=1
ai,j (−1)i+jdet Ac
i,j 
j`eme
det(A) =
p
X
i=1
ai,j (−1)i+jdet Ac
i,j 