Cours algébre

2
3
n
n

R C
∈
a11 a12 . . . a1n
a21 a22 . . . a2n
am1am2. . . amn
aij
m n (m, n)
(aij)
O=
0 0 . . . 0
0 0 . . . 0
0 0 . . . 0
(a1, ..., an)

b1
b2
bm
(aij)aij = 0 i6=j aii = 1 ∀i
I AI =IA =A∀A
I
(aij) (bij)
aij =bij ∀i, j
(m, n) (aij) (bij)
(cij) (m, n)cij =aij +bij i= 1, ..., m
j= 1, ..., n
A= −4 6 3
0 1 2 !B= 5−1 0 3 1 0 !A+
B= 1 5 3
3 2 2 !
A, B C (m, n)
A+B=B+A
(A+B) + C=A+ (B+C)
A+O=O+A=A O
A+ (−A) = O−A= (−aij)

A= (aij)λ∈
λA =
λa11 λa12 . . . λa1n
λa21 λa22 . . . λa2n
λam1λam2. . . λamn
= (λaij)
A= (2 7 8) 3A= (6 21 24)
A, B (m, n)
λ(A+B) = λA +λB
(λ+µ)A=λA +µA
λ(µA) = (λµ)A
1A=A
A= (aij) (m, n)B= (bkl)
(r, p)AB n =r
C= (cil) (m, p)cil =
j=n
X
j=1
aijbjl
A= 3 2 −1
0 4 6 !B=
1 0 2
5 3 1
6 4 2
AB = 3(1) + 2(5) + (−1)(6) 3(0) + 2(3) + (−1)(4) 3(2) + 2(1) + (−1)(2)
0(1) + 4(5) + 6(6) 0(0) + 4(3) + 6(4) 0(2) + 4(1) + 6(2) !
= 726
56 36 16 !
λ(AB) = (λA)B λ ∈
A(BC) = (AB)C
(A+B)C=AC +BC