CPGEdéterminants

n≥2c c =1 2 . . . n −1n
σSnc
Snn≥2σ p
c=a1a2. . . ap
σ◦c◦σ−1p
n(i, j)∈ {1,2, . . . , n}2i6=j σ ∈ Sn
σ τ =i j{i, j}σ
H σ ∈ Snσ(k) + σ(n+ 1 −k) = n+ 1
k∈ {1, . . . , n}
H(Sn,◦)
σ=12345678
35487621σ=12345678 13274856
n∈N∗
σ=1 2 · · · n−1n
n n −1· · · 2 1
σ=123. . . n n + 1 n+ 2 . . . 2n−1 2n
135. . . 2n−1 2 4 . . . 2n−2 2n
n≥2τSn
σ7→ τ◦σSnSn
An
Sn
n≥5
abc a0b0c0Sn
σ
σ◦abc◦σ−1=a0b0c0
F G K
E
f E p F
G q = Id −p
ϕ:E×E→K
ϕ(x, y) = f(p(x))f(q(y)) −f(p(y))f(q(x))
E
ERf E
f2=−Id E
V=x7→ exP(x)P∈Rn[X]
VF(R,R)

D:f7→ f0V
fR C
a, b
∀z∈C, f(z) = az +bz
a b f
A∈ Mn(C)ϕA∈ L(Mn(C))
ϕA(M) = AM
ϕA
A= (ai,j )∈ Mn(R)
∀i∈ {1, . . . , n},|ai,i|>X
j6=i
|ai,j |
A
∀i∈ {1, . . . , n}, ai,i >0
det A > 0
A= (ai,j )∈ Mn(C)A= (ai,j )∈ Mn(C)
det(A) det A
A∈ Mn(C)tA=Adet A∈R
A2n+ 1
det A= 0
A
det(ai,j ) det((−1)i+jai,j ) (ai,j )1≤i,j≤n∈ Mn(K)
A∈ Mn(R)
∀i, j ∈ {1, . . . , n}, ai,j ∈ {1,−1}
2n−1|det A
A∈ Mn(K)C1, . . . , Cn
B
C1−C2, . . . , Cn−1−Cn, Cn−C1
A∈ Mn(R) GLn(Z)A
A∈GLn(Z)|det A|= 1
A, B ∈ Mn(R)
∀k∈ {0,1,...,2n}, A +kB ∈GLn(Z)
det Adet B
A∈ Mn(R)n≥2A1, . . . , AnB∈ Mn(R)
B1, . . . , Bn
Bj=X
i6=j
Ai
det Bdet A

A∈ Mn(C)n≥2X∈ Mn(C)
det(A+X) = det A+ det X
det A= 0 A= 0
A H Mn(R) rg H= 1
det(A+H) det(A−H)≤det A2
A∈ M2n(R)J∈ M2n(R)
∀x∈R,det(A+xJ) = det A
A= (ai,j )∈ Mn(R)
∀(i, j)∈ {1, . . . , n}2, ai,j ≥0∀i∈ {1, . . . , n},
n
X
j=1
ai,j ≤1
|det A| ≤ 1
A∈ Mn(R) det(A2+ In)≥0
0a b
a0c
b c 0
a b c
c a b
b c a
a+b b +c c +a
a2+b2b2+c2c2+a2
a3+b3b3+c3c3+a3
aaaa
a b b b
a b c c
a b c d
a c c b
c a b c
c b a c
b c c a
1 1 1
cos acos bcos c
sin asin bsin c
a1, . . . , an∈Cdet(amax(i,j))
det(max(i, j)) det(min(i, j))
a1, a2, . . . , an∈K
a1a2· · · an
a2
(a1)a1
n∈N∗
S1S1S1· · · S1
S1S2S2· · · S2
S1S2S3· · · S3
S1S2S3· · · Sn
1≤k≤n
Sk=
k
X
i=1
i

A=
a b c d
−b a −d c
−c d a −b
−d−c b a
a, b, c, d ∈R
tA.A det A
a, b, c, d, a0, b0, c0, d0∈Za00, b00, c00, d00 ∈Z
(a2+b2+c2+d2)(a02+b02+c02+d02) = a002+b002+c002+d002
a b c
a2b2c2
a3b3c3
a+b b +c c +a
a2+b2b2+c2c2+a2
a3+b3b3+c3c3+a3
Dn=
1n n −1. . . 2
2 1 3
n−1 1 n
n n −1. . . 2 1
= (−1)n+1 (n+ 1)nn−1
2
n≥2a1, . . . , an
M=ai
aj
+aj
ai1≤i,j≤n
∈ Mn(R)
a6=b λ1, λ2, . . . , λn
∆n(x) = 
λ1+x a +x· · · a+x
b+x λ2+x
a+x
b+x· · · b+x λn+x
[n]
∆n(x)x
∆n(x) ∆n(0)
a1+x(x)
(x)an+x
x, a1, . . . , an
Pn(X) = Xn−X+ 1 n≥2
Pnn z1, . . . , znC
1 + z11· · · 1
1 1 + z2
1
1· · · 1 1 + zn
a, λ1, . . . , λn∈C
H=
a+λ1(a)
(a)a+λn

a1, . . . , an, b1, . . . , bn∈C
ai,j =(ai+bii=j
bj
n≥2 (x1, . . . , xn)n[0 ; π]
Pn=Y
1≤i<j≤n
(cos xj−cos xi)
Mn∈ Mn(R)
mi,j = cos(j−1)xi
mi,j cos xi
det MnPn
n(xk) [0 ; π]
Pn=Y
1≤i<j≤n
(cos xi−cos xj)
Pn
(i, j)∈J1 ; 4K2M∈ M4(R)
mi,j = cos(j−1)xi
mi,j cos xi
det M P4|det M|<24
Dn=
0 1 · · · 1
−1
1
−1· · · −1 0
[n]
Dn=
0 1 · · · 1
1
1
1· · · 1 0
[n]
Dn=
2 1 · · · 1
1 3
1
1· · · 1n+ 1
[n]
(Hn)
Hn=
n
X
k=1
1
k