by 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 by n and n by matrices in response to matttait the determinant of the unique by matrix is not but 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 if one wants the computation of determinants by expansion by minors to work on nonzero by matrices the determinant of the by matrix must be one can also compute how the corresponding linear transformation multiplies dimensional volume the determinant of a block matrix a b is the product of det a and det b for this to hold when a is by and b is a nonsingular matrix one must define det a order of a matrix i don't think this use of order is common in current mathematical literature to me a matrix of order would more likely be a matrix m such that m 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