Matrices

Mn,p(K)

•n, p ∈N∗(n, p)K
Kn p
•n p KMn,p(K)
•A∈Mn,p(K)A ai,j
i j
A= (ai,j )1≤i≤n
1≤j≤p
A= (ai,j )
•A
A=
a1,1a1,2··· a1,p
a2,1a2,2··· a2,p
an,1an,2··· an,p
B=2 3 √2
0 0 π∈M2,3(R)C=0
i∈M2,1(C)
R⊂CMn,p(R)⊂Mn,p(C)
A= (ai,j )1≤i≤n
1≤j≤p
B= (bi,j )1≤i≤n
1≤j≤p
A B
A=B
∀i∈J1, nK∀j∈J1, pKai,j =bi,j .
(1, p)
(n, 1)
A= (ai,j )1≤i≤n
1≤j≤p∈Mn,p(K)AtA AT
Mp,n(K)
tA= (bi,j )1≤i≤p
1≤j≤n∀i∈J1, pK∀j∈J1, nKbi,j =aj,i

B=2 3 √2
0 0 πtB=
2 0
3 0
√2π
•A= (ai,j )B= (bi,j )Mn,p(K)A B
A+B
A+B= (ci,j )1≤i≤n
1≤j≤p∀i∈J1, nK∀j∈J1, pKci,j =ai,j +bi,j .
•A= (ai,j )∈Mn,p(K)λ∈KA
λ λA λA = (λai,j )
•A, B ∈Mn,p(K)A B
KλA +µB λ, µ ∈K
A=1 0 −1
2√2 0 , B =0 0 4 −√213, A +B=
A+√2B=
K(−1)
A B A + (−1)B A −B
Mn,p(K) 0n,p 0
A∈Mn,p(K)
A+ 0n,p = 0n,p +A=A
A∈Mn,p(K)A−A= 0n,p
A, B C Mn,p(K)
•A+B=B+A
•(A+B) + C=A+ (B+C)
A B Mn,p(K)λ, µ K
•(λ+µ)A=λA +µA
•(λµ)A=λ(µA)
•λ(A+B) = λA +µB

A B Mn,p(K)λ, µ K
t(λA +µB) = λtA+µtB.
A= (ai,j )∈Mn,p(K)n p B = (bi,j )∈Mp,q(K)p
q A B AB Mn,q
AB = (ci,j )1≤i≤n
1≤j≤q∀i∈J1, nK∀j∈J1, qKci,j =
p
X
k=1
ai,kbk,j .
A=123
456, B =
0 1
1 0
√2 0
, C =3−1
5 0 
AA AB AC BA BB BC CA CB CC
A∈Mn,p(K)q A.0p,q = 0n,q
0
ab =0=⇒(a= 0 ou b= 0)
R
A=1−1
1−1B=1 1 1 1AB = 02,2
A B C Kλ∈K
•A(B+C) = AB +AC
•(A+B)C=AC +BC
•(λA)B=A(λB) = λ(AB)
•(AB)C=A(BC)

A∈Mn,p(K)B∈Mp,q(K)
t(AB) =tBtA.
(S)
a1,1x1+a1,2x2+··· +a1,p xp=b1
a2,1x1+a2,2x2+··· +a2,p xp=b2
an,1x1+an,2x2+··· +an,p xp=bn
p(x1, . . . , xp)∈Rp(S)X=
x1
xp
AX =B,
A= (ai,j )B=
b1
bn
(S)
Mn,p(K)
Mn,p(K)i∈J1, nKj∈J1, pKEi,j
(i, j)