Fie
M
=
{
1
,
2
,
⋯
,
m
}
,
N
=
{
1
,
2
,
⋯
,
n
}
.
{\displaystyle M=\{1,2,\cdots ,m\},\;N=\{1,2,\cdots ,n\}.}
matrice
(
m
,
n
)
{\displaystyle (m,n)}
A
:
M
×
N
→
C
{\displaystyle A:M\times N\to \mathbb {C} }
A
=
d
e
f
(
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋯
⋯
⋯
⋯
a
n
1
a
n
2
⋯
a
n
n
)
,
{\displaystyle A{\overset {\underset {\mathrm {def} }{}}{=}}{\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\cdots &\cdots &\cdots &\cdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{pmatrix}},}
a
i
j
∈
C
,
i
,
j
=
1
,
n
¯
.
{\displaystyle a_{ij}\in \mathbb {C} ,\;i,j={\overline {1,n}}.}
Se notează prescurtat:
A
=
(
a
i
j
)
i
,
j
∈
M
×
N
.
{\displaystyle A={\big (}a_{ij}{\big )}_{i,j\in M\times N}.}
Mulțimea tuturor matricelor de tip
(
m
,
n
)
{\displaystyle (m,n)}
M
m
n
(
C
)
,
{\displaystyle {\mathcal {M}}_{mn}(\mathbb {C} ),}
m
{\displaystyle m}
n
{\displaystyle n}
m
=
n
,
{\displaystyle m=n,}
matrice pătratică de ordin
n
.
{\displaystyle n.}
Două matrici
A
,
B
∈
M
m
n
(
C
)
{\displaystyle A,B\in {\mathcal {M}}_{mn}(\mathbb {C} )}
egale dacă
a
i
j
=
b
i
j
,
∀
(
i
,
j
)
∈
M
×
N
.
{\displaystyle a_{ij}=b_{ij},\;\forall (i,j)\in M\times N.}
Dându-se matricele
A
,
B
∈
M
m
n
{\displaystyle A,B\in {\mathcal {M}}_{mn}}
A
=
(
a
i
j
)
(
i
,
j
)
∈
M
×
N
{\displaystyle A=(a_{ij})_{(i,j)\in M\times N}}
B
=
(
b
i
j
)
(
i
,
j
)
∈
M
×
N
{\displaystyle B=(b_{ij})_{(i,j)\in M\times N}}
C
=
(
c
i
j
)
(
i
,
j
)
∈
M
×
N
{\displaystyle C=(c_{ij})_{(i,j)\in M\times N}}
c
i
j
=
a
i
j
+
b
i
j
,
∀
(
i
,
j
)
∈
M
×
N
.
{\displaystyle c_{ij}=a_{ij}+b_{ij},\;\forall (i,j)\in M\times N.}
Adunarea matricelor posedă următoarele proprietăți:
(i) asociativitate :
(
A
+
B
)
+
C
=
A
+
(
B
+
C
)
,
∀
A
,
B
,
C
∈
M
m
n
;
{\displaystyle (A+B)+C=A+(B+C),\;\forall A,B,C\in {\mathcal {M}}_{mn};}
(ii) comutativitate :
A
+
B
=
B
+
A
,
∀
A
,
B
∈
M
m
n
;
{\displaystyle A+B=B+A,\;\forall A,B\in {\mathcal {M}}_{mn};}
(iii) existență element neutru
O
∈
M
m
n
{\displaystyle O\in {\mathcal {M}}_{mn}}
A
+
O
=
O
+
A
,
∀
A
∈
M
m
n
.
{\displaystyle A+O=O+A,\;\forall A\in {\mathcal {M}}_{mn}.}
