Find the value of determinant $\begin{vmatrix}1&x&y+z\\1&y&z+x\\1&z&x+y\end{vmatrix}$

$(a)\;0\qquad(b)\;x+y+z\qquad(c)\;1+x+y+z\qquad(d)\;(x-y)(y-z)(z-x)$

$\begin{vmatrix}1&x&y+z\\1&y&z+x\\1&z&x+y\end{vmatrix}$
Apply
$C_3\rightarrow C_3+C_2$
$\begin{vmatrix}1&x&y+z+x\\1&y&z+x+y\\1&z&x+y+z\end{vmatrix}$
Taking out (x+y+z) we have
Since $C_1 and \;C_3$ are identical.
$\Rightarrow \Delta=0$
Hence (a) is the correct answer.
edited Mar 20, 2014