The sum of distinct values of λ for which the system of equations :...

\left|\begin{cases}\lambda - 1 & 3\lambda + 1 & 2\lambda \\ \lambda - 1 & 4\lambda - 2 & \lambda + 3 \\ 2 & 3\lambda + 1 & 3 ( \lambda - 1 )\end{cases}\right| = 0

${R}_{2} \to {R}_{2} - {R}_{1}$

${R}_{3} \to {R}_{3} - {R}_{1}$

\left|$\begin{cases}\lambda - 1 & 3\lambda + 1 & 2\lambda \\ 0 & \lambda - 3 & - \lambda + 3 \\ 3 - \lambda & 0 & \lambda - 3\end{cases}$\right| = 0$</p><p>${C}_{1} \to {C}_{1} + {C}_{3}</p><p>\left|\begin{cases}3\lambda - 1 & 3\lambda + 1 & 2\lambda \\ 3 - \lambda & \lambda - 3 & 3 - \lambda \\ 0 & 0 & \lambda - 3\end{cases}\right| = 0

$⇒ {\left(\lambda - 3\right)}^{2}\left[6\lambda\right] = 0 ⇒ \lambda = 0, 3$

sum of values of $\lambda = 3$