I am trying to figure out at exactly what dimension it is better to use Strassen algorithm rather than the standard multiplication. I know that there is 18 addition and subtraction in Strassen algorithm but I don't know how I must calculate it. This is a homework so I don't want direct answer without any explanation. Any help is appreciated.
The answer to this question depends heavily on the particular details of the computer that you're using.
In modern implementations, conventional matrix multiplication implementation (in the form of highly optimized versions of the BLAS xGEMM function) use blocked algorithms that are carefully tuned to match the cache size of the processor. In comparison, Strassen's algorithm is extremely cache unfriendly, and this makes it difficult to get good performance on contemporary processors.
One recent Arxiv preprint claims that a Strassen like algorithm can give modest performance improvements (up to about 25% faster) than Intel's MKL for matrices of size $N=3000$ or larger on an Intel processor based system:
Austin R. Benson and Grey Ballard. A Framework for Practical Parallel Fast Matrix Multiplication. http://arxiv.org/pdf/1409.2908.pdf
There's another important issue to consider here- Strassen like algorithms are less numerically accurate than conventional blocked matrix-matrix multiplication. In some applications that inaccuracy can cause problems at the application level. Because of this, and because the performance improvements are not that large for reasonably sized matrices, few people make use of Strassen like algorithms in practice.