Question

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

• A. exactly three real values of $t$
• B. exactly real values of $t$
• C. exactly one real value of $t$
• D. infinite number of values of $t$

Answer 1

The correct option is D infinite number of values of $t$

Given system of homogeneous equations is $tx+(t+1)y+(t-1)x=0,(t+1)x+ty+(t+2)z=0,(t-1)x+(t-2)y+tz=0$

For the given system of equations, we have

$A=\left|\begin{array}{ccc}t& t+1& t-1\\ t+1& t& t+2\\ t-1& t-2& t\end{array}\right|$

Solving for $A$, we get

$\Rightarrow A=\left(t\right)(t^2-t^2+4)-(t+1)(t^2+t-t^2-t+2)+(t-1)(t^2-t-2-t^2+t)$

$\Rightarrow A=4t-2t-2-2t+2=0$

The given matrix is a square matrix and we see $A=0$, so it will always

have non-trivial solution for any value of $t$.