A *n* x *n* matrix **A** is invertible if there exists a matrix **A**^{-1} such that **AA**^{-1} = **A**^{-1}A = **I**. This is only a conceivable operation when **A** is a square matrix.

**A**^{-1} exists only when the dimension of the column space of **A**, or the rank of the **A**, is equal to *n*. That is, it only exists when the columns of **A** form a basis for *R*^{n}.

The algorithm described in How to find the inverse of a matrix

will always find the inverse if it exists.