We have a square matrix, that all elements on main diagonal are zero, and other elements are following:
$$a_{i,j}=\begin{cases} 1,&\text{if i+j belongs to Fibonacci numbers,}\\ 0,&\text{if i+j doesn't belong to Fibonacci numbers}.\\ \end{cases}$$
We know that when $n$ is odd the determinant of this matrix is zero.
Now prove that when $n$ is even the determinant of this matrix is $0$ or $1$ or $-1$. (Use induction or other methods).
