I'm having a hard time trying to understand the mathematical proof. I know that if a row or column of a matrix contains constant multiples of a number, say for example $(2,4,6)$ or $(3,6,9)$, its determinant will be $0$. But why?
Edit: By calculating more determinants, I see my mistake. Thank you.
Your statement is not true.
$\begin{bmatrix} 2 & 4 \\ 0 & 1\end{bmatrix}$ doesn't have determinant $0$.
I think what you meant if a row of a matrix is a multiple of another row or a column is a multiple of another column. In that case, you can perform an elementary row operation to eliminate a row or column to obtain a zero row or column. Recall that elementary operation of the form of $cR_i +R_j$ doesn't change the determinant.