Is there a unique identity element for matrices?

[Gear300]

For a set S, there is an identity element e with respect to operation * such that for an element a in S: a*e = e*a = a.

For a matrix B that is m x n, the identity element for matrix multiplication e = I should satisfy IB = BI = B. But for IB, I is m x m, whereas for BI, I is n x n. Doesn't this mean that the two identity elements are different?

 

[HallsofIvy]

If m is not equal to n, then, for b an m x n matrix, b*a maps a m dimensional vector to an n dimensional vector. a*b maps an n dimensional vector to an m dimensional vector. for m not equal to n, b*a= a and a*b= a are both impossible and there is no identity matrix.

 

[Gear300]

If m is not equal to n, then, for b an m x n matrix, b*a maps a m dimensional vector to an n dimensional vector. a*b maps an n dimensional vector to an m dimensional vector. for m not equal to n, b*a= a and a*b= a are both impossible and there is no identity matrix.

So for such cases, would we just say there is a right identity matrix and a left identity matrix? If this was the case, then wouldn't it also imply that the identity mapping is not unique (x² - x² + x = x - x + x)? I was thinking (just right now) that the identity mapping simply depended on the form I(x) = x, in which a mapping is defined through the correspondence between a domain and codomain rather than the process through which the mapping occurs. In the same sense, would we say that an identity matrix is simply a matrix that follows the Kronecker delta form regardless of its dimensions?