  We identify and analyse obstructions to factorisation of integer matrices
into products $N^T N$ or $N^2$ of matrices with rational or integer entries.
The obstructions arise as quadratic forms with integer coefficients and raise
the question of the discrete range of such forms. They are obtained by
considering matrix decompositions over a superalgebra. We further obtain a
formula for the determinant of a square matrix in terms of adjugates of these
matrix decompositions, as well as identifying a $\it co-Latin$ symmetry space.
