If the system of equations $$\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + \lambda z &= 0 \end{aligned}$$ has a unique solution, then $$\lambda$$ is not equal to

Using the unique solution condition $$\Delta \neq 0$$:

$$\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{vmatrix} \neq 0$$

Applying row operation $$R_3 \to R_3 - R_2$$:

$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 0 & 0 & \lambda - 3 \end{vmatrix} \neq 0$$

Expanding along the third row:

$$(\lambda - 3) \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} \neq 0$$

$$(\lambda - 3)(2 - 1) \neq 0 \implies \lambda - 3 \neq 0 \implies \lambda \neq 3$$