Obratna matrika (oznaka
A
−
1
{\displaystyle A^{-1}\,}
A
{\displaystyle A\,}
kvadratne matrike
A
{\displaystyle A\,}
matrika , ki pri množenju z matriko
A
{\displaystyle A\,}
enotsko matriko :
A
⋅
A
−
1
=
I
,
{\displaystyle A\cdot A^{-1}=I\!\,,}
kjer je:
I
{\displaystyle I\,}
n (razsežnosti
n
×
n
{\displaystyle n\times n\,}
A
−
1
{\displaystyle A^{-1}\,}
A
{\displaystyle A\,}
Velja tudi:
A
B
=
B
A
=
I
.
{\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} \!\,.}
Matrike, ki imajo obratno matriko, so obrnljive . Matrika je obrnljiva samo, če je nesingularna. Nekvadratne matrike nimajo obratne matrike (
m
≠
n
{\displaystyle m\neq n\,}
levo in desno obratno matriko . Kadar ima matrika
A
{\displaystyle A\,}
m
×
n
{\displaystyle m\times n\,}
rang enak
n
{\displaystyle n\,}
A
{\displaystyle A\,}
B
.
A
=
I
{\displaystyle B.A=I\,}
B
{\displaystyle B\,}
n
×
m
{\displaystyle n\times m\,}
A
{\displaystyle A\,}
m
{\displaystyle m\,}
B
{\displaystyle B\,}
n
×
m
{\displaystyle n\times m\,}
A
.
B
=
I
{\displaystyle A.B=I\,}
uredi
det
(
A
−
1
)
=
1
det
A
,
{\displaystyle \det(A^{-1})={\frac {1}{\det A}}\!\,,}
kjer je:
det
A
{\displaystyle \det A\,}
determinanta matrike
A
{\displaystyle A\,}
(
A
B
)
−
1
=
B
−
1
A
−
1
{\displaystyle (AB)^{-1}=B^{-1}A^{-1}\!\,}
A
{\displaystyle A\,}
B
{\displaystyle B\,}
(
A
T
)
−
1
=
(
A
−
1
)
T
,
{\displaystyle (A^{\mathrm {T} })^{-1}=(A^{-1})^{\mathrm {T} }\!\,,}
kjer je:
A
T
{\displaystyle A^{\mathrm {T} }\!\,}
transponirana matrika
(
k
A
)
−
1
=
k
−
1
A
−
1
{\displaystyle (kA)^{-1}=k^{-1}A^{-1}\!\,}
k
≠
0
{\displaystyle k\neq 0\,}
(
A
−
1
)
−
1
=
A
{\displaystyle \left(\mathbf {A} ^{-1}\right)^{-1}=\mathbf {A} \!\,}
(
k
A
)
−
1
=
k
−
1
A
−
1
{\displaystyle \left(k\mathbf {A} \right)^{-1}=k^{-1}\mathbf {A} ^{-1}\!\,}
skalar
k
{\displaystyle k\,}
(
A
T
)
−
1
=
(
A
−
1
)
T
{\displaystyle (\mathbf {A} ^{\mathrm {T} })^{-1}=(\mathbf {A} ^{-1})^{\mathrm {T} }\!\,}
za obrnljivi matriki
A
{\displaystyle A\,}
B
{\displaystyle B\,}
n
×
n
{\displaystyle n\times n\,}
(
A
B
)
−
1
=
B
−
1
A
−
1
{\displaystyle \left(\mathbf {AB} \right)^{-1}=\mathbf {B} ^{-1}\mathbf {A} ^{-1}\!\,}
A
1
,
⋯
,
A
k
{\displaystyle A_{1},\cdots ,A_{k}\,}
n
×
n
{\displaystyle n\times n\,}
(
A
1
A
2
⋯
A
k
)
−
1
=
A
k
−
1
A
k
−
1
−
1
⋯
A
1
−
1
{\displaystyle \left(\mathbf {A_{1}A_{2}\cdots A_{k}} \right)^{-1}=\mathbf {A} _{k}^{-1}\mathbf {A} _{k-1}^{-1}\cdots \mathbf {A} _{1}^{-1}\,}
det
(
A
−
1
)
=
det
(
A
)
−
1
.
{\displaystyle \det(\mathbf {A} ^{-1})=\det(\mathbf {A} )^{-1}\!\,.}
uredi
Za določitev obratne matrike se najprej napiše matriko kofaktorjev (adjungirana matrika ):
A
−
1
=
1
|
A
|
(
C
T
)
i
j
=
1
|
A
|
(
C
j
i
)
=
1
|
A
|
(
C
11
C
21
⋯
C
n
1
C
12
C
22
⋯
C
n
2
⋮
⋮
⋱
⋮
C
1
n
C
2
n
⋯
C
n
n
)
,
{\displaystyle \mathbf {A} ^{-1}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} ^{\mathrm {T} }\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} _{ji}\right)={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}{\begin{pmatrix}\mathbf {C} _{11}&\mathbf {C} _{21}&\cdots &\mathbf {C} _{n1}\\\mathbf {C} _{12}&\mathbf {C} _{22}&\cdots &\mathbf {C} _{n2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {C} _{1n}&\mathbf {C} _{2n}&\cdots &\mathbf {C} _{nn}\end{pmatrix}}\!\,,}
kjer je:
|
A
|
{\displaystyle |A|\,}
A
{\displaystyle A\,}
C
i
j
{\displaystyle C_{ij}\,}
C
T
{\displaystyle C^{\mathrm {T} }\,}
Gauss-Jordanova eliminacija
uredi
Gauss-Jordanova eliminacija omogoča ugotoviti, če je neka matrika obrnljiva in določiti tudi obratno matriko. Uporablja se samo za kvadratne matrike. V postopku se najprej dano matriko poveča z enotsko matriko istega reda (dobi se obliko
A
I
{\displaystyle AI\,}
I
A
−
1
{\displaystyle IA^{-1}\,}
Gaussova eliminacijska metoda ).
2
×
2
{\displaystyle 2\times 2}
uredi
Obratno matriko matrike z razsežnostjo
2
×
2
{\displaystyle 2\times 2\,}
A
−
1
=
[
a
b
c
d
]
−
1
=
1
a
d
−
b
c
[
d
−
b
−
c
a
]
.
{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{-1}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\end{bmatrix}}.}
3
×
3
{\displaystyle 3\times 3}
uredi
Obratno matriko v primeru, da se obravnava matriko z razsežnostjo
3
×
3
{\displaystyle 3\times 3\,}
A
−
1
=
[
a
b
c
d
e
f
g
h
k
]
−
1
=
1
Z
[
A
D
G
B
E
H
C
F
K
]
,
{\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\end{bmatrix}}^{-1}={\frac {1}{Z}}{\begin{bmatrix}\,A&\,D&\,G\\\,B&\,E&\,H\\\,C&\,F&\,K\end{bmatrix}}\!\,,}
kjer je:
Z
=
a
(
e
k
−
f
h
)
+
b
(
f
g
−
k
d
)
+
c
(
d
h
−
e
g
)
{\displaystyle Z=a(ek-fh)+b(fg-kd)+c(dh-eg)\,}
Če je
Z
{\displaystyle Z\,}
A
=
(
e
k
−
f
h
)
D
=
(
c
h
−
b
k
)
G
=
(
b
f
−
c
e
)
B
=
(
f
g
−
d
k
)
E
=
(
a
k
−
c
g
)
H
=
(
c
d
−
a
f
)
C
=
(
d
h
−
e
g
)
F
=
(
b
g
−
a
h
)
K
=
(
a
e
−
b
d
)
.
{\displaystyle {\begin{matrix}A=(ek-fh)&D=(ch-bk)&G=(bf-ce)\\B=(fg-dk)&E=(ak-cg)&H=(cd-af)\\C=(dh-eg)&F=(bg-ah)&K=(ae-bd)\!\,.\end{matrix}}}
