Enoncé

n9
n2Mn(R)
n n In
Mn(R)
DnM = (mi,j )16i,j6nMn(R)
(∆1)m1,1mn,n M M
(∆2) M m1,1mn,n
Mn(R)Dn
MDnαM +αIn
Dn
KnMn(R) 1
KnDn
DnMn(R)
(x, y, z)R30x
y z
x y
D2
M2(R)
A = 
3 1 1
0 2 −1
1 1 4 
D3
tM(t) = 
3 1 1 + t
0 2 −1−t
1 1 4 + 2t
M(t)t
tM(t)D3
tM(t)
MM3(R)p
Mp
MM3(R)
0 M M
MM3(R) M3
M3
B= (e1, e2, e3)R3uR3
MB
Ker(u)⊂Ker(u2) Ker(u2)⊂Ker(u3)

Ker(u2) Ker(u3)
u2uii
2
Ker(u) Ker(u2)
(a, b, c, d, e, f)R6M = 
0a b
c0d
e f 0
M3(R)
γ(M) = ac +df +be δ(M) = bcf +ade
M3=γ(M) M +δ(M)I3
Mγ(M) δ(M)
a b d 1
(c, e, f)R3
M
D3
D3
E(X) V(X)
X
n n X
µn(X) = E ([X −E(X)]n).
X
•X 2 3 4
•V(X) 6= 0
X
K(X) = µ4(X)
(µ2(X))2−3 = µ4(X) (V(X))2−3.
X
aP(X = a) = 1 X
Xα β α 6= 0
αX +β
K(αX +β) = K(X).
X [0,1]
X
X
[a, b]a b
a<b
X
Xn n
n µn+2(X) = (n+ 1)µn(X)

X
Xp∈]0,1[
Xp
K(X) p=1
2
Y E(Y2)>E(Y)2
−2
a b X{a, b}
P(X = a) = P(X = b) = 1
2K(X) = −2
X
−2
(X −E(X))2
P(X = x)x
a b
X{a, b}
MK(X) 6M
X
X Y
E(X + Y)4= E X4+ 6 V(X) V(Y) + E Y4
K(X + Y) = V(X)2K(X) + V(Y)2K(Y)
[V(X) + V(Y)]2
X Y
nN∗X1X2Xnn
K n
X
k=1
Xk!=
n
X
k=1
V(Xk)2K(Xk)
n
X
k=1
V(Xk)!2.
X (Xn)n∈N∗
X Sn=
n
X
k=1
Xk
lim
n→+∞K(Sn)=0