The set of homogeneous equations 

$\begin{array}{r}tx+(t+1)y+(t-1)z=0\\ (t+1)x+ty+(t+2)z=0\\ (t-1)x+(t+2)y+tz=0\end{array}$

has non – trivial solutions for :

$$\begin{gathered} \left|\begin{array}{ccc}t& t+1& t-1\\ t+1& t& t+2\\ t-1& t+2& t\end{array}\right|=0 \\ {R}_1 \to R_1-R_{2, }{R}_2 \to R_2-R_3 \\ \left|\begin{array}{ccc}-1& 1& -3\\ 2& -2& 2\\ t-1& t+2& t\end{array}\right|=0 \end{gathered}$$

$-1(-2t-2t-4)-1(2t-2t+2)-3(2t+4+2t-2)=0$

$\text{ D simplifies to }-4(2t+1)=0$