Q:

Determinant property $|c \cdot A| =c^n \cdot |A|$

$$\begin{array}{|ccc|}
x & 2 & 4 \\
x & 1 & 2 \\
x & 4 & 0 \\
\end{array} = x \cdot\begin{array}{|ccc|}
1 & 2 & 4 \\
1 & 1 & 2 \\
1 & 4 & 0 \\
\end{array}$$
Is that property the same as a division? Do I need to say "for $x \neq 0 $"

A:

The determinant mapping is multilinear in the columns; in some books this is part of the definition, in others it is an elementary property. So the equality in your question is true without constraints on $x.$ The equality in the title is then a repeated application of that.

