0-by-0 matrix 

(I'm starting a new subsection to distinguish my comments from the previous unsigned section.)

Because of the connection with linear transformations, it is important to allow 0-by-n and n-by-0 matrices.

In response to MattTait, the determinant of the unique 0-by-0 matrix is not 0, but 1.  There are several ways to see this:
If one defines the determinant as a sum over permutations of products of entries, there is one permutation of the empty set, and the corresponding product is the empty product, which is 1.
If one wants the computation of determinants by expansion by minors to work on nonzero 1-by-1 matrices, the determinant of the 0-by-0 matrix must be 1.
One can also compute how the corresponding linear transformation multiplies 0-dimensional volume.
The determinant of a block matrix 
A 0
0 B
is the product of det(A) and det(B).  For this to hold when A is 0-by-0 and B is a nonsingular matrix, one must define det(A)=1.   

 Order of a matrix 

I don't think this use of "order" is common in current mathematical literature.  To me, a matrix of order 3 would more likely be a matrix M such that M^3 = I.  I think it is better to stick with the standard terminology "dimensions".   

 F^{n x m} 

I think it is more common to write , not , for the space of m-by-n matrices with entries in F.  (I do understand that it is in bijection with linear maps from  to .)