Using
R5517: Cofactor partition for a complex square matrix, we have
\begin{equation}
\text{Det} Y
= \sum_{n = 1}^N Y_{k, n} C_{k, n}
= \sum_{n = 1}^N \lambda A_{k, n} C_{k, n}
= \lambda \sum_{n = 1}^N A_{k, n} C_{k, n}
= \lambda \text{Det} A
\end{equation}
The case for $X$ (where a column is scaled) is proven anologously using the same result. $\square$
