Question:
The sum of distinct values of $\lambda$ for whcih the system of equations
$(\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0$
$(\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0$
$2 x+(3 \lambda+1) y+3(\lambda-1) z=0$
has non-zero solutions, is___________.
Solution:
For non-zero solution, $\Delta=0$
$\Rightarrow\left|\begin{array}{ccc}\lambda-1 & 3 \lambda+1 & 2 \lambda \\ \lambda-1 & 4 \lambda-2 & \lambda+3 \\ 2 & 3 \lambda+1 & 3(\lambda-1)\end{array}\right|=0$
$\Rightarrow 6 \lambda^{3}-36 \lambda^{2}+54 \lambda=0 \Rightarrow 6 \lambda\left[\lambda^{2}-6 \lambda+9\right]=0$
$\Rightarrow \lambda=0, \lambda=3$ [Distinct values]
Then, the sum of distinct values of $\lambda=0+3=3$.