Given system of equations are homogeneous linear equations.

We write in the form AX=0

$ \therefore \left[ \begin{array}{ccc} 3& 1& -\lambda\\ 4& -2& -3\\ 2\lambda& 4& -\lambda \end{array}\right] \left[ \begin{array}{ccc} x\\ y\\ z \end{array}\right] \;=\; \left[ \begin{array}{ccc} 0\\ 0\\ 0 \end{array}\right] \\ \; \\ $

The System has non-trivial solution if the rank of A is less than the number of unknowns, i.e. 3. This requires |A|=0

$ \\ \; \\ \therefore \left| \begin{array}{ccc} 3& 1& -\lambda\\ 4& -2& -3\\ 2\lambda& 4& -\lambda \end{array}\right| \;=\; 0 \\ \; \\ \; \\ \therefore 3[(-2)(-\lambda)-(-3)(4)] \; -1[(4)(-\lambda)-(-3)(2\lambda)] -\lambda[(4)(4)-(-2)(2\lambda)] \;=\; 0 \\ \; \\ \; \\ \therefore 3(2\lambda+12) -1(-4\lambda+6\lambda) -\lambda(16+4\lambda) \;=\; 0 \\ \; \\ \; \\ \therefore 6\lambda+36-2\lambda-16\lambda-4\lambda^2 \;=\;0 \\ \; \\ 4\lambda^2 + 12 \lambda -36\;=\; 0 \\ \lambda^2 + 3 \lambda -9\;=\; 0 \\ \; \\ \; \\ \therefore \lambda \;=\; \dfrac{-3 \pm \sqrt{3^2 \;-\; 4(1)(-9)}}{2} \;=\; \dfrac{-3 \pm \sqrt{45}}{2} \\ \; \\ \; \\ \therefore \lambda \;=\; \dfrac{-3 \pm 3\sqrt{5}}{2} $