Inverse of linear operator

Question

I have came across through a statement that "There exists few operators for which only left inverse exists" and it holds for only right also. But considering the fact that any operator belonging to a finite linear vector space can be represented in matrix form. When we represent the above mentioned special operators in matrix form, it implies that right inverse also exists. How to explain this?

Answer 1

For square matrices, there is no doubt, $AB=I$ implies $detA.detB \neq 0$ so both matrices are invertible. You could have rectangular matrices with a one sided inversion (and obvioulsy no inversion on the other side). In infinite dimension you can have pretty much whatever you want : derivative and integral on the polynomial space for example.

Answer 2

The matrix $\pmatrix{1\\0}$ has a left-inverse, but no right-inverse. On the other hand, the matrix $\pmatrix{0&1}$ has a right-inverse, but no left-inverse.

A square matrix has a left inverse iff it has a right inverse iff it is invertible.