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Sea
A
{\displaystyle A}
c
{\displaystyle c}
A
{\displaystyle A}
c
{\displaystyle c}
c
A
{\displaystyle cA}
A
{\displaystyle A}
c
{\displaystyle c}
Un caso particular es el del producto entre el escalar,-1, y una matriz cualquiera
A
{\displaystyle A}
(
−
1
)
A
{\displaystyle (-1)A}
−
A
{\displaystyle -A}
−
A
{\displaystyle -A}
A
{\displaystyle A}
i
)
c
(
A
+
B
)
=
c
A
+
c
B
.
{\displaystyle i)\ c(A+B)=cA+cB.\,}
i
i
)
(
a
+
b
)
C
=
a
C
+
b
C
.
{\displaystyle ii)\ (a+b)C=aC+bC.\,}
i
i
i
)
(
a
b
)
C
=
a
(
b
C
)
=
b
(
a
C
)
.
{\displaystyle iii)\ (ab)C=a(bC)=b(aC).\,}
i
v
)
1
A
=
A
.
{\displaystyle iv)\ 1A=A.\,}
v
)
(
−
1
)
A
=
−
A
.
{\displaystyle v)\ (-1)A=-A.\,}
La fórmula general para la operación de cA es:
Sea A una matriz cualquiera y c un escalar
A
=
[
a
11
a
12
.
.
a
1
n
a
21
a
22
.
.
a
2
n
a
31
a
32
.
.
a
3
n
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
.
.
a
m
n
]
,
c
{\displaystyle A=\left[{\begin{array}{ccccc}a_{11}&a_{12}&.&.&a_{1n}\\a_{21}&a_{22}&.&.&a_{2n}\\a_{31}&a_{32}&.&.&a_{3n}\\.&.&.&.&.\\.&.&.&.&.\\a_{m1}&a_{m2}&.&.&a_{mn}\\\end{array}}\right],c}
A
{\displaystyle A}
c
{\displaystyle c}
c
A
{\displaystyle cA}
c
A
=
[
c
a
11
c
a
12
.
.
c
a
1
n
c
a
21
c
a
22
.
.
c
a
2
n
c
a
31
c
a
32
.
.
c
a
3
n
.
.
.
.
.
.
.
.
.
.
c
a
m
1
c
a
m
2
.
.
c
a
m
n
]
{\displaystyle cA=\left[{\begin{array}{ccccc}ca_{11}&ca_{12}&.&.&ca_{1n}\\ca_{21}&ca_{22}&.&.&ca_{2n}\\ca_{31}&ca_{32}&.&.&ca_{3n}\\.&.&.&.&.\\.&.&.&.&.\\ca_{m1}&ca_{m2}&.&.&ca_{mn}\\\end{array}}\right]}
Donde el escalar, c, se multiplica por cada uno de los elementos de la matriz
A
{\displaystyle A}
Sean
A
=
[
−
5
6
1
3
]
,
c
=
−
1
{\displaystyle A=\left[{\begin{array}{cc}-5&6\\1&3\end{array}}\right],c=-1}
El producto cA es igual a
c
A
=
−
1
[
−
5
6
1
3
]
=
[
−
1
×
−
5
−
1
×
6
−
1
×
1
−
1
×
3
]
=
[
5
−
6
−
1
−
3
]
{\displaystyle cA=-1\left[{\begin{array}{cc}-5&6\\1&3\end{array}}\right]={\begin{bmatrix}-1\times -5&-1\times 6\\-1\times 1&-1\times 3\end{bmatrix}}={\begin{bmatrix}5&-6\\-1&-3\end{bmatrix}}}
A
=
[
−
3
2
−
5
−
1
0
−
2
3
−
4
1
]
,
c
=
2
{\displaystyle A=\left[{\begin{array}{ccc}\!-3&\,2&\!-5\\\!-1&\,0&\!-2\\\,3&\!-4&\,1\end{array}}\right],c=2}
c
A
{\displaystyle cA}
c
A
=
2
[
−
4
5
−
5
−
5
0
−
5
3
−
10
0
]
=
[
2
×
−
3
2
×
2
2
×
−
5
2
×
−
1
2
×
0
2
×
−
2
2
×
3
2
×
−
4
2
×
1
]
=
[
−
6
4
−
10
−
2
0
−
4
6
−
8
2
]
{\displaystyle cA=2\left[{\begin{array}{ccc}\!-4&\,5&\!-5\\\!-5&\,0&\!-5\\\,3&\!-10&\,0\end{array}}\right]=\left[{\begin{array}{ccc}\ 2\times -3&2\times 2&2\times -5\\\ 2\times -1&2\times 0&2\times -2\\\ 2\times 3&2\times -4&2\times 1\end{array}}\right]=\left[{\begin{array}{ccc}\!-6&\,4&\!-10\\\!-2&\,0&\!-4\\\,6&\!-8&\,2\end{array}}\right]}
George Nakos, David Joyner: Álgebra Lineal con Aplicaciones . Estados Unidos:Us Naval Academy, 1999.
666 p. ISBN 968-7529-86-5 .
responsable: Jesús David Ramos Rengifo 20082005074
JesusDavidRamos 23:10 7 mar 2009 (UTC)jesus david ramos rengifo 20082005074
![{\displaystyle A=\left[{\begin{array}{ccccc}a_{11}&a_{12}&.&.&a_{1n}\\a_{21}&a_{22}&.&.&a_{2n}\\a_{31}&a_{32}&.&.&a_{3n}\\.&.&.&.&.\\.&.&.&.&.\\a_{m1}&a_{m2}&.&.&a_{mn}\\\end{array}}\right],c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4baf0e41a93f4eb7de7b3a8087096ab5df7db5)
![{\displaystyle cA=\left[{\begin{array}{ccccc}ca_{11}&ca_{12}&.&.&ca_{1n}\\ca_{21}&ca_{22}&.&.&ca_{2n}\\ca_{31}&ca_{32}&.&.&ca_{3n}\\.&.&.&.&.\\.&.&.&.&.\\ca_{m1}&ca_{m2}&.&.&ca_{mn}\\\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1b2f6c348160f18a91ee2e29973b47181a47f4)
![{\displaystyle A=\left[{\begin{array}{cc}-5&6\\1&3\end{array}}\right],c=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca34a7cf421d2d65704f2b2052b38b793cca4d5)
![{\displaystyle cA=-1\left[{\begin{array}{cc}-5&6\\1&3\end{array}}\right]={\begin{bmatrix}-1\times -5&-1\times 6\\-1\times 1&-1\times 3\end{bmatrix}}={\begin{bmatrix}5&-6\\-1&-3\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeeb9320283a5a2b7a60b37dc8d5a2bd60e7d9c)
![{\displaystyle A=\left[{\begin{array}{ccc}\!-3&\,2&\!-5\\\!-1&\,0&\!-2\\\,3&\!-4&\,1\end{array}}\right],c=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d90eeee5077ccdea329e518499c78db45e2c56a)
![{\displaystyle cA=2\left[{\begin{array}{ccc}\!-4&\,5&\!-5\\\!-5&\,0&\!-5\\\,3&\!-10&\,0\end{array}}\right]=\left[{\begin{array}{ccc}\ 2\times -3&2\times 2&2\times -5\\\ 2\times -1&2\times 0&2\times -2\\\ 2\times 3&2\times -4&2\times 1\end{array}}\right]=\left[{\begin{array}{ccc}\!-6&\,4&\!-10\\\!-2&\,0&\!-4\\\,6&\!-8&\,2\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d171917fe3aacbae2f9b92c6d8c0fc0746d455)