# × × pm B = B = BA⋅ =

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```Multiplicaci&oacute;n de matrices
Definici&oacute;n:
Sean
El
A =[ a ij ] una matriz de m &times; n
B =[ bij ] una matriz de n &times; p
producto
de
las
matrices
m &times; p definida por A ⋅ B =[ cik ]
A y B es
la
matriz
n
donde
c = ∑ a ⋅b
ik
j =1
ij
Ejemplos:
1)
Sean
entonces
2)
1 2

 4
A = 3
A⋅B
y
0 − 1
8 

B = 5
1 ⋅ 0 + 2 ⋅ 5 1 ⋅ (−1) + 2 ⋅ 8  10
=
=
3 ⋅ 0 + 4 ⋅ 5 3 ⋅ (−1) + 4 ⋅ 8 20
2 1 
3 7 
A

=
Sean
4 6
y
15 
29
3 1 9

 7 0
B = 5
entonces
 2 ⋅ 3 + 1 ⋅ 5 2 ⋅ 1 + 1 ⋅ 7 2 ⋅ 9 + 1 ⋅ 0  11 9 18 


A ⋅ B = 3 ⋅ 3 + 7 ⋅ 5 3 ⋅ 1 + 7 ⋅ 7 3 ⋅ 9 + 7 ⋅ 0  = 44 52 27
4 ⋅ 3 + 6 ⋅ 5 4 ⋅ 1 + 6 ⋅ 7 4 ⋅ 9 + 6 ⋅ 0 42 46 36 
jk
de
Observaciones:
1)
2)
3)
En general, si
Si A ⋅ B y
distinto.
A⋅B
A ⋅ B est&aacute; definida B ⋅ A no lo est&aacute;.
B ⋅ A est&aacute;n definidas, en general son de tama&ntilde;o
B ⋅ A est&aacute;n ambas definidas y adem&aacute;s tienen el
mismo tama&ntilde;o ( A y B son cuadradas de igual tama&ntilde;o) , en
Si
y
general son distintas , o sea, el producto de matrices NO es
conmutativo).
Definici&oacute;n:
A ⋅ B = B ⋅ A entonces se dice que A
A y B son matrices conmutativas.
Si
y
B conmutan o que
Ejemplo:
Sean
A
A⋅B
1 1
= 0 1


=B⋅
A
y
2 5 

 2
B = 0
2 7 
= 0 2 entones


A
y
B conmutan.
α ∈ ℜ y para toda
B y C de n &times; p :
Para todo
matrices
1)
matriz
A
A ⋅(B + C ) = A ⋅ B + A ⋅C
de
m&times;n
y
2)
α ( A ⋅ B ) = A ⋅ (α B ) = (α A ) ⋅ B
A⋅ In =In ⋅ A
de orden n .
=
Para toda matriz
p&times;r
A
de
A
, donde
m&times;n
In
B
,
de
n&times; p
A ⋅ (B ⋅C ) = ( A ⋅ B) ⋅C
Para toda matriz
(A ⋅ B)
t
A
de
m&times;n
y
B
de
n&times; p
t
t
⋅
B
A
=
Definici&oacute;n:
Sea
A
Se define
A
0
=
In
y
A
m
n , m ∈ℵ
=
A
m −1
⋅A
,
,
C de
Ejemplo:
A
Sea
A
1
A
2
A
3
=
11 − 25
=4 −9 


A
=
A⋅A
=
2
A ⋅A
11 − 25
⋅
−
9


=4
11 − 25 21 − 50
 4 − 9  =  8 − 19 


 
21 − 50 11 − 25 31 − 75
 ⋅  4 − 9  = 12 − 29
−
19

 

 
= 8
```