Proving complexity of computing product of matrices
I'm afraid you can't.  As I mentioned in my response to your previous question:


  The complexity of a problem is the running time of the fastest algorithm for that problem.


There exist algorithms for this problem whose running time is much less than $N^3$.  As I described in my previous answer, there are much more efficient algorithms for matrix inversion (and for matrix multiplication).  I linked to algorithms for both.

Therefore, you cannot prove that the complexity is $\frac83 N^3 + \dots$ flops -- because that isn't true.

Given your pair of questions, and since you said you're not sure where to start, I would suggest you start by reviewing the following material here on this site:


What is the difference between an algorithm, a language and a problem?
How to come up with the runtime of algorithms?
How does one know which notation of time complexity analysis to use?
Is there a system behind the magic of algorithm analysis?


Then, read some material on fast algorithms for matrix multiplication and other matrix operations.  That'll be a starting point, at least.

If what you want is to understand the running time of LU-factorization of a matrix, that's a different question.  If that's your question, you should probably research that, and then post a new question, showing what research you've done.