left inverse and right inverse of matrices

Question

If $A \in R^{m×n}$ has a left inverse, $A^{T}$ has a right inverse. How I can prove this property? I hope someone can help me.

Answer 1

Let $L$ be the left inverse of $A$. Then

$$LA=I.$$

Now transpose the equation $A^TL^T=I^T$.

Answer 2

If $B$ is a left inverse of $A$, then $B^T$ is a right inverse of $A^T$, because$$A^TB^T=(BA)^T=\operatorname{Id}^T=\operatorname{Id}.$$