将矩阵 \(A\) 写成如下形式:
\[A=\begin{pmatrix} -2a_1 & 0 & \cdots & 0 & 0 \\ 0 & -2a_2 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & -2a_{n-1} & 0 \\ 0 & 0 & \cdots & 0 & -2a_n \end{pmatrix}\]
\[+\begin{pmatrix} a_1 & 1 \\ a_2 & 1 \\ \vdots & \vdots \\ a_{n-1} & 1 \\ a_n & 1 \end{pmatrix}\cdot I_2^{-1}\cdot\begin{pmatrix} 1 & 1 & \cdots & 1 & 1 \\ a_1 & a_2 & \cdots & a_{n-1} & a_n \end{pmatrix}.\]
由降阶公式可得
\[|A|=\begin{vmatrix} -2a_1 & 0 & \cdots & 0 & 0 \\ 0 & -2a_2 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & -2a_{n-1} & 0 \\ 0 & 0 & \cdots & 0 & -2a_n \end{vmatrix}\cdot\Bigg|I_2+\begin{pmatrix} 1 & 1 & \cdots & 1 & 1 \\ a_1 & a_2 & \cdots & a_{n-1} & a_n \end{pmatrix}\begin{pmatrix} -2a_1 & 0 & \cdots & 0 & 0 \\ 0 & -2a_2 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & -2a_{n-1} & 0 \\ 0 & 0 & \cdots & 0 & -2a_n \end{pmatrix}^{-1}\begin{pmatrix} a_1 & 1 \\ a_2 & 1 \\ \vdots & \vdots \\ a_{n-1} & 1 \\ a_n & 1 \end{pmatrix}\Bigg|\]
\[=(-2)^n\prod_{i=1}^na_i\begin{vmatrix} 1-\frac{n}{2} & -\frac{1}{2}\sum_{i=1}^n\frac{1}{a_i} \\ -\frac{1}{2}\sum_{i=1}^na_i & 1-\frac{n}{2} \end{vmatrix}\]
\[=(-2)^{n-2}\prod_{i=1}^na_i\bigg((n-2)^2-\Big(\sum_{i=1}^na_i\Big)\Big(\sum_{i=1}^n\frac{1}{a_i}\Big)\bigg). \quad\Box\]