Conditions under which $BA = I_{n}$, where $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{n\times m}$ Let $A\in\mathbb{C}^{m\times n}$ . I want to to know what conditions can I apply  on the matrix $B\in\mathbb{C}^{n\times m}$ such that product  $BA = I_{n}$ or matrix $B$ is the left inverse of the matrix $A$.
Please help me. Thanks for the help.

I think that, as @OwenSizemore correctly suggested, if $n\leq m$ and A is a full rank matrix, then we can think of some Moore-Penrose Pseudoinverse type of solution. 
If not, then I'm not sure if this is even possible. Given a vector $\mathbf{v}$ such that $\mathbf{Ax=v}$ we can't really get $\mathbf{x}$ since given a solution $\mathbf{x_0}$ to this equation, $\mathbf{x_0+n}$ is also a solution ($\mathbf{n}$ is any vector in the nullspace of $\mathbf{A}$), right? 
