The determinant is linear in one row or column with every other row or column fixed. For example, $\det(u+v,c_2,\ldots)=\det(u,c_2,\ldots)+\det(v,c_2,\ldots)$. Scaling all $n$ columns by $c$ therefore gives $\det(cA)=c^n\det A$, not $c\det A$ in general.
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