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

Question

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

Answer 1

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.

Answer 2

No, it is not division, it is "factoring". If x= 0 then obviously both sides are 0 so the equation is still true.