page 169 note § The question of the validity of this theorem for n = 1 is one of some interest, though it has little or no practical importance. While, according to this theorem, the pure one-element determinant |a| vanishes, we are assured with equal certainty from the theory of elimination from linear equations, that the same pure determinant should have the value a. The contradiction arises from the fact that our theory of determinants is concerned only with determinants of definite dimensionality. A determinant of order one is of indeterminate dimensions, corresponding to the geometrical fact that a point could be considered as the shrinking limit of a piece of continuum of any number of dimensions. Hence if we are given a one-element determinant |a|, and if we are told in addition that it is a p-dimensional determinant with q Scott suffixes, we would logically be in a position to evaluate it ; namely |a| = a or 0, according as p – q is even or odd.
The following slightly different point of view may perhaps be considered preferable. The concept of determinant as a number associated with an ordered spatial array, is based essentially on the notion of suffix. Now a suffix is something which possesses necessarily a field of variation. But if the order n is equal to one, there is no field of variation and therefore, strictly speaking, there are no suffixes (unless convention steps in and says that there are). Hence determinants of order unity are determinants of zero dimensions, and therefore may be legitimately considered to be meaningless.
As this case does not occur in practice, it is immaterial which of these two views we adopt.
