Showing a matrix pattern is invertable for certain dimensions

Question

The problem is finding for which dimension n the nxn square matrix below is invertible. I used rref and found the pattern followed a distinct pattern. It seemes to be linearly dependent only when n is a combination of n=5+3k (eg: 5, 8, 11, etc) . However I don't know really why or how I can explain it. Any tips?

Answer 1

Let $D_n = \det(B_n)$. Try to establish a recurrence relation relating $D_n$ to terms $D_{n - 1}$ and $D_{n - 2}$. Reveal the spoiler below to see the answer.

$D_n = D_{n - 1} - D_{n - 2}$ with $D_1 = 1, D_0 = 0$.

When you solve the recurrence, you should find that the matrix is singular whenever $n = 3k + 2$, where $k \in \mathbb N$.