Namespaces
Variants
Actions

Difference between revisions of "Inverse matrix"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
−
''of a square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523901.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523902.png" />''
+
{{TEX|done}}
 +
''of a square matrix $A$ over a field $k$''
  
−
The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523903.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523904.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523905.png" /> is the identity matrix. Invertibility of a matrix is equivalent to its being non-singular (see [[Non-singular matrix|Non-singular matrix]]). For the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523906.png" />, the inverse matrix is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523907.png" /> where
+
The matrix $A^{-1}$ for which $AA^{-1}=A^{-1}A=E$, where $E$ is the identity matrix. Invertibility of a matrix is equivalent to its being non-singular (see [[Non-singular matrix|Non-singular matrix]]). For the matrix $A=\|\alpha_{ij}\|$, the inverse matrix is $A^{-1}=\|\gamma_{ij}\|$ where
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523908.png" /></td> </tr></table>
+
$$\gamma_{ij}=\frac{A_{ji}}{\det A},$$
  
−
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i0523909.png" /> is the [[Cofactor|cofactor]] of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052390/i05239010.png" />. For methods of computing the inverse of a matrix see [[Inversion of a matrix|Inversion of a matrix]].
+
where $A_{ij}$ is the [[Cofactor|cofactor]] of the element $\alpha_{ij}$. For methods of computing the inverse of a matrix see [[Inversion of a matrix|Inversion of a matrix]].

Revision as of 10:38, 6 September 2014

of a square matrix $A$ over a field $k$

The matrix $A^{-1}$ for which $AA^{-1}=A^{-1}A=E$, where $E$ is the identity matrix. Invertibility of a matrix is equivalent to its being non-singular (see Non-singular matrix). For the matrix $A=\|\alpha_{ij}\|$, the inverse matrix is $A^{-1}=\|\gamma_{ij}\|$ where

$$\gamma_{ij}=\frac{A_{ji}}{\det A},$$

where $A_{ij}$ is the cofactor of the element $\alpha_{ij}$. For methods of computing the inverse of a matrix see Inversion of a matrix.

How to Cite This Entry:
Inverse matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_matrix&oldid=33255
Retrieved from "https://encyclopediaofmath.org/index.php?title=Inverse_matrix&oldid=33255"