Inverse Matrix ( If there are two different left inverse for a matrix then the matrix has no right inverse)

inverselinear algebra

My question is very short:

Let $A$ $n$ by $m$ matrix.

  • If there are two different left inverse for A then the matrix has no right inverse, right?
  • If $A$ is invertible, then must $A$ be $n=m$?

Best Answer

  • Right. If $A$ has a left inverse, it is injective. If it had a right inverso too, then it would be surjective and therefore a linear isomorphism. But a linear isomorphism only has one left inverse.
  • Right, once again. Just apply the rank-nullity theorem, for instance.