We have \begin{equation} \begin{split} \text{Det} (\lambda A) & = \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N \lambda A_{n, \pi(n)} \right) \\ & = \sum_{\pi \in S_N} \lambda^N \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right) \\ & = \lambda^N \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right) \\ & = \lambda^N \text{Det} A \\ \end{split} \end{equation}