Find $\det(C_n)$ where $c_{ij} = 1$ unless $i-j=\pm 1$

Question

Find $\det(C_n)$ where $c_{ij} = 1$ unless $i-j=\pm 1$.

The original problem from the quiz,

Let $C_n$ be the $n$ by $n$ matrix whose entries are all ones, except for zeros directly below and above the main diagonal; for example, $$ C_5 = \begin{bmatrix} 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 \end{bmatrix} $$

Find $\det(C_n)$

Any insights are welcome.

Answer 1

Hint: Apply row operations. $C_5$, for example, has the same determinant as $$ \pmatrix{ 1 &0 &1&1&1\\ 0&1&0&1&1\\ 0&0&0&-1&0 \\ 1&1&0&1&0\\ 1&1&1&0&1 } $$