  We consider LU and QR matrix decompositions using exact computations. We show
that fraction-free Gauss--Bareiss reduction leads to triangular matrices having
a non-trivial number of common row factors. We identify two types of common
factors: systematic and statistical. Systematic factors depend on the reduction
process, independent of the data, while statistical factors depend on the
specific data. We relate the existence of row factors in the LU decomposition
to factors appearing in the Smith--Jacobson normal form of the matrix. For
statistical factors, we identify some of the mechanisms that create them and
give estimates of the frequency of their occurrence. Similar observations apply
to the common factors in a fraction-free QR decomposition. Our conclusions are
tested experimentally.
