# Prove the determinant of this matrix

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).

-
Welcome to MSE! Please verify I got everything correct. What have you tried? Where are you confused? Regards – Amzoti Feb 11 at 17:13
how can i prove that ? – Zoha Shams Feb 11 at 17:22
Hint: write out the 2x2 and 3x3 cases. Prove your statements for those. Next, use induction to prove the general case. Regards. – Amzoti Feb 11 at 17:24
i know that i should use induction, i ask you how can use induction?could u prove it? – Zoha Shams Feb 11 at 17:26
Posted a few days ago to MO, mathoverflow.net/questions/121243/… --- best to check the progress there before spending too much time on it. – Gerry Myerson Feb 12 at 5:34