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.