If the equations $$ \begin{aligned} & k x+(k+1) y+(k-1) z=0, \\ & (k-1) x+(k+2) y+k z=0 \text { and } \\ & (k+1) x+k y+(k+2) z=0 \end{aligned} $$ have a non trivial solution, then the sum of all the possible values of $k$ is

• A

  0
• Correct answer
  B

  $-\frac{1}{2}$
• C

  $\frac{1}{2}$
• D

  1

Hence, the correct answer is option: 'B'.