A
−
1
=
1
|
A
|
(
C
i
j
)
T
=
1
|
A
|
(
C
j
i
)
=
1
|
A
|
(
C
11
C
21
⋯
C
j
1
C
12
C
22
⋯
C
j
2
⋮
⋮
⋱
⋮
C
1
i
C
2
i
⋯
C
j
i
)
{\displaystyle \mathbf {A} ^{-1}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} _{ij}\right)^{\mathrm {T} }={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} _{j1}\\\mathbf {C} _{12}&\mathbf {C} _{22}&\cdots &\mathbf {C} _{j2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {C} _{1i}&\mathbf {C} _{2i}&\cdots &\mathbf {C} _{ji}\\\end{pmatrix}}}
sin
x
+
ln
y
a
2
+
2
{\displaystyle \sin x+\ln ya^{2+2}}
∙
1
2
{\displaystyle \bullet {\frac {1}{2}}}
2
3
→
‖
∣
{\displaystyle {\tfrac {2}{3}}\to \|\mid }
⇒
‡
‖
‖
∄
{\displaystyle \Rightarrow \ddagger \lVert \rVert \nexists }
∥
{\displaystyle \parallel }
a
2
a
2
{\displaystyle a^{2}a_{2}}
{\displaystyle \ \,}
x
y
z
v
{\displaystyle {\begin{matrix}x&y\\z&v\end{matrix}}}
|
x
y
z
v
|
{\displaystyle {\begin{vmatrix}x&y\\z&v\end{vmatrix}}}
‖
x
y
z
v
‖
{\displaystyle {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}}
[
x
y
z
v
]
{\displaystyle {\begin{bmatrix}x&y\\z&v\end{bmatrix}}}
r
i
g
h
t
c
e
n
t
e
r
l
e
f
t
r
c
l
{\displaystyle {\begin{array}{rcl}right¢er&left\\r&c&l\end{array}}}
{
3
x
+
5
y
+
z
=
a
7
x
−
2
y
+
4
z
=
b
−
6
x
+
3
y
+
2
z
=
c
{\displaystyle {\begin{cases}3x+5y+z=a\\7x-2y+4z=b\\-6x+3y+2z=c\end{cases}}}
[
s
t
u
f
f
]
{\displaystyle \left[stuff\right]}
General Matrix Inversion
edit
A matrix that is square and not singular has a single inverse. The transpose of the inverse of the transpose of the matrix is also equal to the inverse. The formulae
A
−
1
T
=
A
−
1
T
{\displaystyle A^{-1^{T}}=A^{-1^{T}}}
A
T
−
1
T
=
A
−
1
{\displaystyle A^{T^{-1^{T}}}=A^{-1}}
[
2
0
1
|
0
−
1
2
|
1
1
0
|
1
0
0
0
1
0
0
0
1
1
0
0
|
0
1
0
|
0
0
1
|
]
{\displaystyle \left[{\cfrac {{\begin{matrix}2&0&1&|\\0&-1&2&|\\1&1&0&|\end{matrix}}\,{\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}}}{{\begin{matrix}1&0&0&|\\0&1&0&|\\0&0&1&|\end{matrix}}{\begin{matrix}\quad \quad \,&\end{matrix}}}}\right]}
Must modify the layout to make the type of display I need. Hmmmmm... I'll have to think about this a bit.
rand.
Invertible Matrix
rand huso (talk ) 21:38, 3 February 2008 (UTC)
![{\displaystyle \left[stuff\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8189e9efe80dd7a115c11a7f442575970c8e03)
![{\displaystyle \left[{\cfrac {{\begin{matrix}2&0&1&|\\0&-1&2&|\\1&1&0&|\end{matrix}}\,{\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}}}{{\begin{matrix}1&0&0&|\\0&1&0&|\\0&0&1&|\end{matrix}}{\begin{matrix}\quad \quad \,&\end{matrix}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863c48cb542d483f5ad2a0006bd691552dbd250d)