I've been trying to solve this problem but I stucked. Let $A = [a_{ij}]$ with size $ 2011 \times 2011$ , and given the condition below
\begin{equation}a_{ij}= \begin{cases} (-1)^{|i-j|}, & \text{if}\ i \neq j \\ 2, & \text{if} \ i = j \end{cases} \end{equation} Find $\det(A)$. I was thinking to either form block matrices or do row operation, but it became a mess. Please help me to figure this out.
Hint We can write $A$ as $$A = I_{2011} + {\bf x} {\bf x}^\top ,$$ where ${\bf x} = (1, -1, 1, -1, \ldots, -1, 1)^\top$. Now apply the Matrix Determinant Lemma.