Definition:Inverse Matrix

Definition
Let $\left({R, +, \circ}\right)$ be a ring with unity.

Let $\mathcal M_R \left({n}\right)$ be the $n \times n$ matrix space over $R$.

Let $\mathbf A$ be an element of the ring $\left({\mathcal M_R \left({n}\right), +, \times}\right)$.

Let $\mathbf A$ be invertible in $\mathcal M_R \left({n}\right)$.

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

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

Also see

 * Product Inverse in Ring is Unique