I'm given a signal $$x(t)=\cos(200\pi t)$$ which is sampled at $t=n/400$ instance, for $n=0,1,2,..,7$. I have to comment on $X$, the 8 point discrete Fourier transform.
Just multiplying by the DFT matrix with input signal, gives me $$X=[0,0,4,0,0,0,4,0].$$ But I'm unable to understand why exactly X[2] and X[6] are non-zero.
The frequency step corresponding to the given sampling rate and no. of samples is $$\Delta f=\frac{1}{no. of samples\times sampling\space rate}$$ $$=\frac{1}{8\left(\frac{1}{400}\right)}$$ $$=50$$ Now, your DFT matrix consists of the values of DFT coefficients at $f=n\Delta f ; n=0,1,...,7$
Since the frequency of your input signal is $\frac{200\pi}{2\pi}=100=2\Delta f$, you get your first peak at $f=100$, i.e. corresponding to $n=2$. Its value is obviously $\frac{N}{2}, N$= no. of samples.
Now, the second half of the coefficient matrix is the complex conjugate of the first half (from Nyquist Sampling Theorem), i.e. $$c_{k}=c_{N-k}^{\#},$$$$k=1,2,...,6; N=8$$
$c_{k}$ is the $k^{th}$ element of the DFT matrix (numbering starts from $0$), hence the second non-zero entry.