Hi,
I could indeed tell you that I have a PhD in Mathematics and a degree in Electronics, and point you to my publications, where I use Fourier series. This is all true, but I understand that you have been deceived before, hence, I will just give you some clues to answer the third question:
The rank of a matrix depends on the independence of its lines and columns. If we reduce your matrix, using pivots, we end up with (1 2 3; 0 10 14; 0 0 6.16). None of the lines are null, and hence the rank is 3.
I'm not a fake. You can contact me to discuss this further.
Best Regards,
Isabelle