Let $z \in \mathbb{C}$ be a complex number. Using the definition for the eigenvalue sequence, we have
\begin{equation}
f(z)
: = \text{Det}(z I_N - A)
= \prod_{n = 1}^N (z - \lambda_n)
\end{equation}
Therefore, using result
R5569: Determinant of reflected complex matrix, we have
\begin{equation}
(-1)^N \text{Det} A
= \text{Det} (- A)
= f(0)
= \prod_{n = 1}^N (0 - \lambda_n)
= (-1)^N \prod_{n = 1}^N \lambda_n
\end{equation}
Diving both sides by $(-1)^N$, we conclude
\begin{equation}
\text{Det} A
= \prod_{n = 1}^N \lambda_n
\end{equation}
$\square$
