Fie A un inel și
m
,
n
∈
N
∗
.
{\displaystyle m, n \in \mathbb N^*.}
α
=
(
α
11
α
12
⋯
α
1
n
α
21
α
22
⋯
α
2
n
α
m
1
α
m
2
⋯
α
m
n
)
{\displaystyle \alpha = \begin{pmatrix} \alpha_{11} & \alpha_{12} & \cdots & \alpha_{1n} \\ \alpha_{21} & \alpha_{22} & \cdots & \alpha_{2n} \\ \alpha_{m1} & \alpha_{m2} & \cdots & \alpha_{mn} \end{pmatrix} \!}
cu m linii și n coloane, format din elemente ale lui A se numește matrice cu m linii și n coloane ; convenim să notăm a astfel de matrice și sub formă mai condensată:
α
=
(
α
i
j
)
1
≤
j
≤
n
1
≤
i
≤
m
.
{\displaystyle \alpha = (\alpha_{ij})_{\overset{1 \le i \le m}{1 \le j \le n}}. \!}
m
=
n
{\displaystyle m=n \!}
M
m
,
n
(
A
)
=
M
n
(
A
)
;
{\displaystyle M_{m,n}(A) =M_n(A); \!}
M
n
(
A
)
{\displaystyle M_n(A) \!}
pătratică de ordin n .
α
,
β
∈
M
m
,
n
,
α
=
(
α
i
j
)
1
≤
j
≤
n
1
≤
i
≤
m
{\displaystyle \alpha, \beta \in M_{m,n}, \; \; \alpha = (\alpha_{ij})_{\overset{1 \le i \le m}{1 \le j \le n}} \!}
β
=
(
β
i
j
)
1
≤
j
≤
n
1
≤
i
≤
m
{\displaystyle \beta = (\beta_{ij})_{\overset{1 \le i \le m}{1 \le j \le n}} \!}
α
+
β
=
(
α
i
j
+
β
i
j
)
1
≤
j
≤
n
1
≤
i
≤
m
.
{\displaystyle \alpha + \beta = (\alpha_{ij} + \beta_{ij})_{\overset{1 \le i \le m}{1 \le j \le n}}. \!}
Asociativitatea adunării matricelor este imediată, elementul neutru este matrice
O
m
,
n
{\displaystyle O_{m,n} \!}
M
m
,
n
(
A
)
{\displaystyle M_{m,n}(A) \!}
α
{\displaystyle \alpha \!}
−
α
=
(
−
α
i
j
)
1
≤
j
≤
n
1
≤
i
≤
m
,
{\displaystyle - \alpha =(- \alpha_{ij})_{\overset{1 \le i \le m}{1 \le j \le n}}, \!}
(
M
m
,
n
(
A
)
,
+
)
{\displaystyle (M_{m, n}(A), +) \!}
Pentru
m
,
n
,
p
∈
N
∗
,
α
∈
M
m
,
n
(
A
)
,
β
∈
M
n
,
p
(
A
)
,
α
=
(
α
i
j
)
1
≤
j
≤
n
1
≤
i
≤
m
,
β
=
(
β
i
j
)
1
≤
k
≤
p
1
≤
j
≤
n
{\displaystyle m,n, p \in \mathbb N^*, \; \alpha \in M_{m,n}(A), \; \beta \in M_{n,p} (A), \; \alpha = (\alpha_{ij})_{\overset{1 \le i \le m}{1 \le j \le n}}, \; \beta = (\beta_{ij})_{\overset{1 \le j \le n}{1 \le k \le p}} \!}
α
β
=
(
γ
i
k
)
1
≤
k
≤
p
1
≤
i
≤
m
,
{\displaystyle \alpha \beta = (\gamma_{ik})_{\overset{1 \le i \le m}{1 \le k \le p}}, \!}
γ
i
k
=
∑
j
=
1
α
i
j
β
j
k
{\displaystyle \gamma_{ik} = \sum_{j=1} \alpha_{ij} \beta_{jk} \!}
1
≤
j
≤
m
{\displaystyle 1 \le j \le m \!}
1
≤
k
≤
p
.
{\displaystyle 1 \le k \le p. \!}
În mod evident,
α
β
∈
M
m
,
p
(
A
)
.
{\displaystyle \alpha \beta \in M_{m,p}(A). \!}
Să demonstrăm că dacă
m
,
n
,
p
,
q
∈
N
∗
{\displaystyle m, n, p ,q \in \mathbb N^* \!}
α
∈
M
m
,
n
(
A
)
,
β
∈
M
n
,
p
(
A
)
,
γ
∈
M
p
,
q
(
A
)
,
{\displaystyle \alpha \in M_{m, n}(A), \; \beta \in M_{n, p}(A), \; \gamma \in M_{p, q}(A), \!}
(
α
β
)
γ
=
α
(
β
γ
)
.
{\displaystyle (\alpha \beta) \gamma = \alpha(\beta \gamma). \!}
α
β
=
(
δ
i
k
)
1
≤
k
≤
p
1
≤
i
≤
m
,
{\displaystyle \alpha \beta = (\delta_{ik})_{\overset{1 \le i \le m}{1 \le k \le p}}, \!}
δ
i
k
=
∑
j
=
1
n
α
i
j
β
j
k
{\displaystyle \delta_{ik} = \sum_{j=1}^n \alpha_{ij}\beta_{jk} \!}
(
α
β
)
γ
=
(
ε
i
t
)
1
≤
t
≤
q
1
≤
i
≤
m
{\displaystyle (\alpha \beta) \gamma = (\varepsilon_{it})_{\overset{1 \le i \le m}{1 \le t \le q}} \!}
ε
i
t
=
∑
k
=
1
p
δ
i
k
γ
k
t
=
∑
k
=
1
p
(
∑
j
=
1
n
α
i
j
β
j
k
)
γ
k
t
=
∑
k
=
1
p
∑
j
=
1
n
α
i
j
β
j
k
γ
k
t
.
{\displaystyle \varepsilon_{it}= \sum_{k=1}^p \delta_{ik}\gamma_{kt} = \sum_{k=1}^p (\sum_{j=1}^n \alpha_{ij} \beta_{jk})\gamma_{kt} = \sum_{k=1}^p \sum_{j=1}^n \alpha_{ij}\beta_{jk}\gamma_{kt}. \!}
β
γ
=
(
δ
j
t
′
)
1
≤
t
≤
q
1
≤
j
≤
n
{\displaystyle \beta \gamma = (\delta'_{jt})_{\overset{1 \le j \le n}{1 \le t \le q}} \!}
δ
j
t
′
=
∑
k
=
1
p
β
j
k
γ
k
t
{\displaystyle \delta'_{jt} = \sum_{k=1}^p \beta_{jk} \gamma_{kt} \!}
α
(
β
γ
)
=
(
ε
i
t
′
)
1
≤
t
≤
q
1
≤
i
≤
m
,
{\displaystyle \alpha (\beta \gamma) = (\varepsilon'_{it})_{\overset{1 \le i \le m}{1 \le t \le q}} , \!}
ε
i
t
′
=
∑
j
=
1
n
α
i
j
δ
j
t
′
=
∑
j
=
1
n
α
i
j
∑
k
=
1
p
β
j
k
γ
k
t
=
∑
j
=
1
n
∑
k
=
1
p
α
i
j
β
j
k
γ
k
t
,
{\displaystyle \varepsilon'_{it} = \sum_{j=1}^n \alpha_{ij} \delta'_{jt} = \sum_{j=1}^n \alpha_{ij} \sum_{k=1}^p \beta_{jk} \gamma_{kt} = \sum_{j=1}^n \sum_{k=1}^p \alpha_{ij} \beta_{jk} \gamma_{kt}, \!}
(
α
β
)
γ
=
α
(
β
γ
)
.
{\displaystyle (\alpha \beta) \gamma = \alpha(\beta \gamma). \!}
Ținând cont de distributivitatea înmulțirii de pe A față de adunare, deducem imediat că dacă
α
∈
M
m
,
n
(
A
)
{\displaystyle \alpha \in M_{m,n} (A) \!}
β
,
γ
∈
M
n
,
p
(
A
)
{\displaystyle \beta, \gamma \in M_{n, p}(A) \!}
α
(
β
+
γ
)
=
α
β
+
α
γ
{\displaystyle \alpha (\beta + \gamma) = \alpha \beta + \alpha \gamma \!}
α
,
β
∈
M
m
,
n
(
A
)
{\displaystyle \alpha, \beta \in M_{m, n} (A) \!}
γ
∈
M
n
,
p
(
A
)
{\displaystyle \gamma \in M_{n, p} (A) \!}
(
α
+
β
)
γ
=
α
γ
+
β
γ
.
{\displaystyle (\alpha + \beta) \gamma = \alpha \gamma + \beta \gamma. \!}
n
∈
N
,
n
≥
2
,
{\displaystyle n \in \mathbb N, \; n \ge 2, \!}
(
M
n
(
A
)
,
+
,
⋅
)
{\displaystyle (M_n(A), + , \cdot) \!}
inel (denumit inelul matricelor pătratice de ordin n cu elemente din A ) .
Dacă inelul A este unitar, atunci și inelul
(
M
n
(
A
)
,
+
,
⋅
)
{\displaystyle (M_n(A), + , \cdot) \!}
I
n
{\displaystyle I_n \!}
Să remarcăm faptul că în general, chiar dacă A este comutativ,
M
n
(
A
)
{\displaystyle M_n(A) \!}
Resurse [ ] 
