It is not so clear from your question, but I suppose that what you call a ''dilatation'' is an homotetic transformation in an affine space.
In this case let the two transformations be: $$ d_1(M)=S+\lambda_1 \overrightarrow {SM} $$ $$ d_2(M)=S+\lambda_2 \overrightarrow {SM} $$ where $S$ is the common fixed point and $\lambda_{1,2}$ the two scale factors, than we have: $$ d_2\left(d_1(M)\right)=S+\lambda_2 \overrightarrow {Sd_1(M)} $$ where: $$ \overrightarrow {Sd_1(M)}=\lambda_1\overrightarrow {SM} $$
so: $$ d_2\left(d_1(M)\right)=S+\lambda_2 \lambda_1\overrightarrow {SM} $$ is an homotetic transformation with the same fixed point $S$ and scale factor $\lambda_3=\lambda_1\lambda_2$
The commutativity is easily proved from this.