Definition:Inverse Matrix

Definition
Let $\struct {R, +, \circ}$ be a ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\mathcal M_R} n, +, \times}$.

Let $\mathbf A$ be invertible in $\map {\mathcal M_R} n$.

Then the inverse of $\mathbf A$ is defined as:
 * $\mathbf A^{-1} \in \map {\mathcal M_R} n: \mathbf A \mathbf A^{-1} = \mathbf I_n = \mathbf A^{-1} \mathbf A$

where $\mathbf I_n$ denotes the unit matrix of order $n$.

Also see

 * Product Inverse in Ring is Unique