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.