Let $F_n$ denote the $n$th Fibonacci number, adopting the convention $F_1=1$, $F_2=1$ and so on. Consider the $n\times n$ matrix defined by
I have the following conjecture:
Conjecture. For all integers $n\geq3$, $\det\mathbf M_n=0$.
I have used some Python code to test this conjecture for $n$ up to $9$, but I cannot go further. Note that $\det\mathbf M_1=\det\mathbf M_2=1$. Due to the elementary nature of this problem I have to assume that it has been discussed before, perhaps even on this site. But I couldn't find any reference on it, by Googling or searching here. Can someone shed light onto whether the conjecture is true, and a proof of it if so?