How to work out the inverse matrix $A^{-1}$ ?

Question

Suppose A is a matrix over some ring R (might be non-commutative). How to work out the inverse matrix $A^{-1}$?

Answer 1

Consider $A$ as a homomorphism of a free $R$-module. If this homomorphism is invertible, then you can build $A^{-1}$ which is a matrix of the inverse homomorphism.

Answer 2

Should you be so lucky that $R$ is commutative, then iff the determinant of $A$ is invertible in $R$, you can take the adjugate matrix and multiply it with the inverse of the determinant squared, see the section named "inverse". This might be possible to apply to some matrices over some non-commutative rings too, I have little experience with those.