I have proved that if a square $n$ x $n$ matrix $A$ has a right and left inverse, then these are equal and form an inverse matrix of $A$.
However I'm interested in the following implication:
Suppose a left inverse $B$ of a square $n$ x $n$ matrix $A$ exist. Does this imply that a right inverse $C$ of $A$ exist ?
Also, if this is true - is the implication also true in the case of a right inverse $B$ ?