For $\mathbb{Z}/2 \mathbb{Z}$ (and other finite fields of characteristic $2$) there is a specialized library for compting with matrices over that structure.
It is called M4RI ; on this website in particular under further reading one can find various text related to this.
In particular, a paper by developpers of the library:
Martin Albrecht, Gregory Bard, William Hart. Algorithm 898: Efficient Multiplication of Dense Matrices over GF(2). ACM Transactions on Mathematical Software 2010.
Preprint at http://arxiv.org/abs/0811.1714.
Yet, the main point there, as far as my understanding goes, is that modolu $2$ arithmetic can be done very efficiently on a computer and the point is to really optimize the methods to exploit this.
Let me also say some things in part already in the comments:
I (also) believe for multiplication if one counts say just number of multiplications over the base-structure and takes this as a measure of 'goodness' it does not matter whether one is over $\mathbb{Z}$ or modulo $n$. This also seems to be in line with the fact that in M4RI $O(n^{\log_2 7})$ is mentioned and one also has this for integers.
Yet, for certain computations related to integer matrices, it is useful to pass to modular arithmetic and then go back, yet (only) due to the fact that the arithemtic over the base structure (mod $n$ vs. integers) can be much faster; in particular, if the integers involved are large or (perhaps more importantly) can grow large in the process.
For example, to compute determinants of integers matrices a strategy can be to on the one hand compute a bound on the determinant (e.g., using Hadamard's bound) and
on the other hand to compute the determinant modulo many primes.
As then combinining the modulo $p$ pieces of information on the determinant, one knows the determinant modulo such a large modulus that actually there remains only a unique integer satisfying both the congruence conditions and the size condition (implied by the bound).
Things like this are for example discussed in
H. Cohen, A course in computational Algebraic Number Theory, Springer GTM 138.