Uniqueness of identity elements for rectangular matrices

[Bipolarity]

Let A be the set of $n \times n$ matrices. Then the identity element of this set under matrix multiplication is the identity matrix and it is unique. The proof follows from the monoidal properties of multiplication of square matrices.

But if the matrix is not square, the left and right identities are not equivalent; they are both identity matrices, but have a different size.

How do you know that the left-identity is unique, and that the right-identity is unique?
So given an $m \times n$ matrix A, how do you know that the only matrix satisfying $AI = A$ for all A is the $n \times n$ identity matrix?

Is this even true? Could I possibly find multiple right-identity elements?

BiP

 

[micromass]

Let $B$ be an identity element. Let $A$ be the $m\times n$ matrix with a 1 on the $i$th row and the $j$th column and zero everywhere else. You know that $AB = A$. This gives a condition on $B$. Which one? What if you vary $i$ and $j$?