The sum of distinct values of for which the system of equations 

$\begin{array}{l}(\lambda -1)x+(3\lambda +1)y+2\lambda z=0\\ (\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0\\ 2x+(3\lambda +1)y+3(\lambda -1)z=0\end{array}$

has non zero solutions, is................

$\begin{array}{ll}\left|\begin{array}{cccc}\lambda & -1& 3\lambda +1& 2\lambda \\ \lambda & -1& 4\lambda -2& \lambda +3\\ 2& 3\lambda +1& 3(\lambda -1)& \end{array}\right|=0& \\ \Rightarrow & \left|\begin{array}{ccc}2\lambda & 10\lambda & 6\lambda \\ \lambda -1& 4\lambda -2& \lambda +3\\ 2& 3\lambda +1& 3\lambda -3\end{array}\right|=0\left(R_1+R_2+R_3\right)\\ \Rightarrow & 2\lambda \left|\begin{array}{ccc}1& 5& 3\\ \lambda -1& 4\lambda -2& \lambda +3\\ 2& 3\lambda +1& 3\lambda -3\end{array}\right|=0\\ & 2\lambda \left|\begin{array}{ccc}1& 0& 0\\ \lambda -1& -\lambda +3& -2\lambda +6\\ 2& 3\lambda -9& 3\lambda -9\end{array}\right|=0\\ C_2-5C_1& C_3-3C_1\end{array}$

$\begin{array}{l}\Rightarrow \lambda (\lambda -3{)}^2\left|\begin{array}{ccc}1& 0& 0\\ \lambda -1& -1& -2\\ 2& 3& 3\end{array}\right|=0\\ \Rightarrow \lambda (\lambda -3{)}^2=0\\ \Rightarrow \lambda =0\text{ or }3\\ \text{ Sum }=3\end{array}$