PrecalculusSuperchallengeSystems, Matrices, and Determinants

Systems, Matrices, and Determinants · Superchallenge problem

There is a constant $k$ such that, for all real numbers $a$ , $b$ , $c$ ,

\det\begin{bmatrix}a+b&b+c&c+a\\b+c&c+a&a+b\\c+a&a+b&b+c\end{bmatrix}=k\cdot\det\begin{bmatrix}a&b&c\\b&c&a\\c&a&b\end{bmatrix}.

Find $k$ .