Wikipedia, Entziklopedia askea
Alderantzizko matrizea matrizea da, beste matrize batekin biderkatuta, emaitza gisa unitate matrizea ematen duena. Matrize karratuek bakarrik dute alderantziko matrizea, baina ez denek. A matrizearen alderantziko matrizea
A
−
1
{\displaystyle A^{-1}}
adierazten da eta bien arteko biderketak baldintza hau betetzen du:
A
⋅
A
−
1
=
A
−
1
⋅
A
=
I
n
{\displaystyle A\cdot A^{-1}=A^{-1}\cdot A=I_{n}}
, non
I
n
{\displaystyle I_{n}}
n ordenako unitate matrizea den.
Determinante ez nulua daukan matrize bat emanik:
A
−
1
=
[
a
b
c
d
]
−
1
=
1
det
(
A
)
[
d
−
b
−
c
a
]
=
1
a
d
−
b
c
[
d
−
b
−
c
a
]
{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}}
Definituta dago
det
(
A
)
=
a
d
−
b
c
≠
0
{\displaystyle {\det(\mathbf {A} )}=ad-bc\neq 0}
baldin bada. Adibidez:
[
2
1
5
3
]
↦
[
2
1
5
3
]
−
1
=
[
3
−
1
−
5
2
]
,
{\displaystyle {\begin{bmatrix}2&1\\5&3\end{bmatrix}}\mapsto {\begin{bmatrix}2&1\\5&3\end{bmatrix}}^{-1}={\begin{bmatrix}3&-1\\-5&2\end{bmatrix}},}
[
2
1
5
3
]
[
3
−
1
−
5
2
]
=
[
1
0
0
1
]
{\displaystyle {\begin{bmatrix}2&1\\5&3\end{bmatrix}}{\begin{bmatrix}3&-1\\-5&2\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}
baita
A
−
1
=
[
a
b
c
d
e
f
g
h
i
]
−
1
=
1
det
(
A
)
[
A
B
C
D
E
F
G
H
I
]
T
=
1
det
(
A
)
[
A
D
G
B
E
H
C
F
I
]
{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,I\\\end{bmatrix}}^{T}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,D&\,G\\\,B&\,E&\,H\\\,C&\,F&\,I\\\end{bmatrix}}}
Definituta dago
det
(
A
)
≠
0
{\displaystyle {\det(\mathbf {A} )}\neq 0}
baldin bada.
A
=
(
e
i
−
f
h
)
D
=
−
(
b
i
−
c
h
)
G
=
(
b
f
−
c
e
)
B
=
−
(
d
i
−
f
g
)
E
=
(
a
i
−
c
g
)
H
=
−
(
a
f
−
c
d
)
C
=
(
d
h
−
e
g
)
F
=
−
(
a
h
−
b
g
)
I
=
(
a
e
−
b
d
)
{\displaystyle {\begin{matrix}A=(ei-fh)&D=-(bi-ch)&G=(bf-ce)\\B=-(di-fg)&E=(ai-cg)&H=-(af-cd)\\C=(dh-eg)&F=-(ah-bg)&I=(ae-bd)\\\end{matrix}}}