Let $n\geq1$ be an integer. Take the matrix $M(n)$, with entries, $M\_{i,j}(n)=\sin\left(\frac{(i+j)\pi}2\right)$ if $i\neq j$ and $M\_{i,i}(n)=x\_i$.


I wish to ask (this question has been modified from my previous post):



> 
> **QUESTION.** Is there any interpretation/meaning to the matrix $M(n)$ and its determinant?
> $$\det M(n)=x\_1x\_2\cdots x\_n-\sum\_{\substack{1\leq i\_1<\cdots<i\_{n-2}\leq n\\
>  \binom{n+1}2-(i\_1+\cdots+i\_{n-2})\,\equiv\, 1\,\, \text{mod}\, 2}}x\_{i\_1}\cdots x\_{i\_{n-2}}.$$
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> 


**Example.** If $n=4$ then (updated using Fedor's rewrite)
\begin{align\*}
\det\begin{pmatrix} x\_1&-1&0&1 \\ -1&x\_2&1&0 \\ 0&1&x\_3&-1 \\ 1&0&-1&x\_4 \end{pmatrix}
&=x\_1x\_2x\_3x\_4-x\_1x\_2-x\_1x\_4-x\_2x\_3-x\_3x\_4 \\
&=x\_1x\_2x\_3x\_4\left(1-(x\_1^{-1}+x\_3^{-1})(x\_2^{-1}+x\_4^{-1})\right).
\end{align\*}



Interpret this matrix and its determinant
Well,
\begin{align\*}
\sum\_{\substack{1\leq i\_1<\cdots<i\_{n-2}\leq n\\
 \binom{n+1}2-(i\_1+\cdots+i\_{n-2})\,\equiv\, 1\,\, \text{mod}\, 2}}x\_{i\_1}\cdots x\_{i\_{n-2}}
&=x\_1\cdots x\_n\cdot \left(\sum\_{i+j\,\text{is odd}}(x\_ix\_j)^{-1}\right)\\
&=x\_1\cdots x\_n(x\_1^{-1}+x\_3^{-1}+\ldots)(x\_2^{-1}+x\_4^{-1}+\ldots).
\end{align\*}


