<s>[INST] Prove determinant is zero If  
$M = \begin{vmatrix}
        1 & a & b+c \\
        1 & b & a+c \\
        1 & c & a+b \\
        \end{vmatrix}$
Show that M = 0 WITHOUT expanding the determinant.
I have tried row operations and haven't had much success.  Any tips?   
 [/INST] $$\begin{vmatrix}1&a&b+c\\
1&b&a+c\\
1&c&a+b\end{vmatrix}\stackrel{R_2-R_1\,,\,\,R_3-R_1}\longrightarrow\begin{vmatrix}1&a&b+c\\
0&b-a&a-b\\
0&c-a&a-c\end{vmatrix}= $$
$${}$$
$$=(a-b)(a-c)\begin{vmatrix}1&a&b+c\\
0&-1&1\\
0&-1&1\end{vmatrix}$$
and clearly we reached two equal rows
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