determinant

[Lena]

\( a, b, c, d \in C. \)Aratati ca
\( \det \begin{pmatrix} b + c + d & a & a & a \\
b & a + c + d & b & b \\
c & c & a + b + d & c \\
d & d & d & a + b + c \\
\end{pmatrix} = 4\sum abc(a + b + c) \)

[Marius Mainea]

\( LHS=\sum\left|\begin{array}S&a&a&a\\0&-b&b&b\\0&c&-c&c\\0&d&d&-d\end{array}\right|+\left|\begin{array}{cc}-a&a&a&a\\b&-b&b&b\\c&c&-c&c\\d&d&d&-d\end{array}\right|=4\sum Sbcd-16abcd=RHS \)

unde \( S=a+b+c+d \)