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