# Inverse matrix

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 15A09 [MSN][ZBL]

*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.

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Inverse matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Inverse_matrix&oldid=36931