The system of homogeneous equations tx+(t+1)y+(t−1)z=0,t+1x+ty+(t+2)z=0,t−1x+t+2y+tz=0 has a non-trivial solution for

The condition for the system of equations to have a non-trivial solution is|A|=0⇒tt+1t-1t+1tt+2t-1t+2t=0, Apply R3→R3-R2;R2→R2-R1⇒tt+1t-11-13-22-2=0, Apply: C1→C1+C2⇒2t+1t+1t-10-1302-2=0⇒(2t+1)2-6=0⇒t=-1/2.