The value of $\begin{vmatrix}2y_1 z_2&y_1 z_2+y_2z_1&y_1z_3+y_3z_1\\y_1 z_2+y_2z_1&2y_2z_2&y_2z_3+y_3z_2\\y_1z_3+y_3z_1&y_2z_3+y_3z_2&2y_3z_3\end{vmatrix}$, is

0

We have,

$\begin{vmatrix}2y_1 z_2&y_1 z_2+y_2z_1&y_1z_3+y_3z_1\\y_1 z_2+y_2z_1&2y_2z_2&y_2z_3+y_3z_2\\y_1z_3+y_3z_1&y_2z_3+y_3z_2&2y_3z_3\end{vmatrix}$

$=\begin{vmatrix}y_1 z_1+y_1 z_1&y_1 z_2+y_2z_1&y_1z_3+y_3z_1\\y_1 z_2+y_2z_1&y_2z_2+y_2z_2&y_2z_3+y_3z_2\\y_1z_3+y_3z_1&y_2z_3+y_3z_2&y_3z_3+y_3z_3\end{vmatrix}$

$=\begin{vmatrix}y_1&z_1&0\\y_2&z_2&0\\y_3&z_3&0\end{vmatrix}\begin{vmatrix}z_1&y_1&0\\z_2&y_2&0\\z_3&y_3&0\end{vmatrix}=0×0=0$