<s>[INST] How should I calculate the determinant? $\left|\begin{array}{cccc}1&a&b&c+d\\1&b&c&a+d\\1&c&d&a+b\\1&d&a&b+c\end{array}\right|=
\left|\begin{array}{cccc}1&a&b&c\\1&b&c&a\\1&c&d&a\\1&d&a&b\end{array}\right|+
\left|\begin{array}{cccc}1&a&b&d\\1&b&c&d\\1&c&d&b\\1&d&a&c\end{array}\right|$
I tried to calculate the determinant but I couldn't do it after separating the determinant by the property.  How should I calculate it?  
 [/INST] $${\begin{vmatrix}1&a & b &c\\1 &b &c &d \\1 &c &d &a\\1&d &a &b &\end{vmatrix}}=-{\begin{vmatrix}1&a & b &d\\1 &b &c &a \\1 &c &d &b\\1&d &a &c &\end{vmatrix}}$$
because the second determinant is the same as the first by cyclically permuting the columns $2,3$ and $4$, an cyclically permuting the four rows.
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