Inverse matrix

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.