Prove:$$\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
I tried to use the Laplace expansion, but it seems useless.
2026-03-30 05:28:43.1774848523
Proving $\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
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Hint:
$C_1'=C_1+C_2+C_3$
$$\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}$$
$$=\begin{vmatrix}a+b+c&b&c\\b+c+a&c&a\\c+a+b&a&b\end{vmatrix}$$
$$=(a+b+c)\begin{vmatrix}1&b&c\\1&c&a\\1&a&b\end{vmatrix}$$
Now either expand or
use $R_2'=R_2-R_1, R_3'=R_3-R_1$