[Math] Prove determinant is zero

Question

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?

Answer 1

Hint: Add the second column to the third, and use the fact that if the columns of a matrix are linearly dependent, then the matrix has determinant zero.