why should I scale the fft using 1/N? I am writing a report, and my advisor asked me to explain why I scale the fft by a factor 1/N (where N is the length of the array). 

I used to use the scaling convention of multipling the fft by the time increment (dt), this convention was good for me, because it ensures the check of the Parseval theorem. Unfortunately I had discussion with one of my advisors, because, since this convention does not give you the right amplitude, he thinks it is wrong. As I have read online, there are not right convention or wrong convention. If I use the factor 1/N the amplitude is checked, if I use the factor dt then the parseval identity is preserved.
I have 2 question now:


Why, doing the fft, I cannot have both: amplitude checked and energy checked?
I have already demonstrated (in my report) that if I scale the fft by a factor 1/N, I obtain the right amplitude, since the first value of my fft is equal to the time average of my function. Now I would like to show with formulas why this convention gives me the right amplitude? I have already searched online and here on the forum, but I did not find any good answer that explain every passage.
OK, let us go for a 2-point DFT. It should be noted that, depending on the software used, scaling can be different, and ought to be checked. 
The standard unscaled version does multiply the input vector by:

$$\left[\begin{matrix} 1 &amp; 1\\1 &amp; -1\end{matrix}\right]$$

We have two more options: preserve the average, thus we need:

$$\frac{1}{{2}}\left[\begin{matrix} 1 &amp; 1\\1 &amp; -1\end{matrix}\right]$$

as $\frac{1}{{2}}\left[\begin{matrix} 1 &amp; 1\\1 &amp; -1\end{matrix}\right]\left[\begin{matrix} x_1\\x_2\end{matrix}\right]$ has the same average as $\left[\begin{matrix} x_1\\x_2\end{matrix}\right]$, or 

$$\frac{1}{\sqrt{2}}\left[\begin{matrix} 1 &amp; 1\\1 &amp; -1\end{matrix}\right]$$

to preserve energy (thus orthogonality) in both domains. More generally, you have three standard options:


avoid scaling
scale by $\frac{1}{\sqrt{N}}$
or by $\frac{1}{{N}}$, 


depending on the invariance purpose. But you can scale columns with other quantities, as needed for your computational purpose. 
The following code aims at showing that the $1/\sqrt{N}$ scaling could work (if I am not too tired) to preserve Parseval-Plancherel or Rayleigh's energy theorem.

nSample = 2^1 ; % One can change the 2^* to other powers
data = randn(nSample,1) ; 
dataFFT = fft(data)/sqrt(nSample); 
ratioParseval = (norm(data)-norm(dataFFT))/norm(data)