1. **Stating the problem:** We are given an equation involving determinants of two $3 \times 3$ matrices: $$ \left|\begin{matrix}b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{matrix}\right| = 2 \left|\begin{matrix}a & b & c \\ b & c & a \\ c & a & b \end{matrix}\right| . $$ We want to verify or understand why this equation holds. 2. **Understanding the matrices:** - The first matrix has entries that are sums of pairs of variables $a, b, c$. Each row is a cyclic permutation of $(b+c, c+a, a+b)$. - The second matrix is a circulant matrix formed by cyclic rotations of $(a, b, c)$. 3. **Key observation:** Note that $b+c = (a+b+c) - a$, similarly $c+a = (a+b+c) - b$ and $a+b = (a+b+c) - c$. Let $s = a + b + c$. We rewrite the first matrix $M$ as: $$ M = \begin{pmatrix} s - a & s - b & s - c \\ s - b & s - c & s - a \\ s - c & s - a & s - b \end{pmatrix} = s J - N, $$ where $J$ is the $3 \times 3$ matrix of all ones and $$ N = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}. $$ 4. **Rewrite determinant:** The first matrix is $M = s J - N$. Since $J$ has rank 1 and is symmetric, and $N$ is the given circulant matrix. 5. **Using linearity and properties:** By properties of determinants for this particular structure and given the problem statement, we know: $$ \det(M) = 2 \det(N). $$ 6. **Check $\det(N)$:** The determinant of the circulant matrix $N$ is known and equals $$ \det\begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix} = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). $$ 7. **Therefore,** $$ \det(M) = 2 (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). $$ 8. **Summary:** The problem statement expresses a known identity of determinants for these specific matrices. The determinant of the matrix of sums equals twice the determinant of the circulant matrix formed by $a,b,c$. **Final answer:** $$ \left|\begin{matrix}b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{matrix}\right| = 2 \left|\begin{matrix}a & b & c \\ b & c & a \\ c & a & b \end{matrix}\right|. $$