设 $s_k=\lambda_1^k+\lambda_2^k+\cdots+\lambda_n^k(k=1,2, \cdots)$, 求证：
$$
\left|\begin{array}{ccccc}
n & s_1 & s_2 & \cdots & s_{n-1} \\
s_1 & s_2 & s_3 & \cdots & s_n \\
\cdots \\
s_{n-1} & s_n & s_{n+1} & \cdots & s_{2 n-2}
\end{array}\right|=\prod_{1 < j < i < n}\left(\lambda_i-\lambda_j\right)^2 .
$$