The number of real values of t such that the system of homogeneous equations aligned t x+(t+1) y+(t-1) z & =0 (t+1) x+t y+(t+2) z & =0 (t-1) x+(t+2) y+t z & =0 aligned has non-trivial solutions is

Step-by-step solution

Given, aligned & t x+(t+1) y+(t-1) z=0 & (t+1) x+t y+(t+2) z=0 & (t-1) x+(t+2) y+t z=0 aligned Here, Coefficient matrix, A= [ array ccc t & t+1 & t-1 t+1 & t & t+2 t-1 & t+2 & t array ] If |A|=0 , then system of equations has non-trivial solution and it has infinite solutions. |A|= | array ccc t & t+1 & t-1 t+1 & t & t+2 t-1 & t+2 & t array |=0 Apply operation R₂ R₂-R₁, R₃ R₃-R₁ , we get = | array ccc t & t+1 & t-1 1 & -1 & 3 -1 & 1 & 1 array |=0 Apply C₂ C₂-C₁, C₃ C₃-C₁ , we get | array crr t & 1 & -1 1 & -2 & 2 -