Thank you Carl for your response to my query.
In the 674 dimensional example, the maximal degree for the numerator and denominator entries p(x)/q(x) in the matrices over $Q(x)$ is 5. The maximum density of non-zero entries is approximately 0.8%.
If I change the basis so that the matrices are over Z[x], the maximum degree becomes 1. Actually, the original problem involves either starting with a collection rational matrices over Q(x), which act on a vector space V over Q(x), and doing a lot of matrix multiplications and additions in order to construct a basis over the integers Z[x] for an integral subspace of V, or vice versa.
Here is the result of the 34 second calculation, run on a slightly older computer this time
memory used=4.58GiB, alloc change=-236.99MiB, cpu time=46.82s, real time=47.56s, gc time=22.09s
This time, I ran it on a slightly older and slower i7-2620 M CPU @2.70 GHZ, which probably accounts for the extra time.
The structure of the matrices varies a lot, depending on whether they are over $Q(x)$ or $Z[x]$. However, in the multiplication computed above, all non-zero entries happen to be close to the diagonal.