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.

Answer 2

sum all columns to the right and then factored

Answer 3

$$\begin{vmatrix}1&a&b+c\\ 1&b&a+c\\ 1&c&a+b\end{vmatrix}\stackrel{R_2-R_1\,,\,\,R_3-R_1}\longrightarrow\begin{vmatrix}1&a&b+c\\ 0&b-a&a-b\\ 0&c-a&a-c\end{vmatrix}= $$

$${}$$

$$=(a-b)(a-c)\begin{vmatrix}1&a&b+c\\ 0&-1&1\\ 0&-1&1\end{vmatrix}$$

and clearly we reached two equal rows

Answer 4

If you add the second column to the third one, the resulting column is a multiple of the first one.