For what value of $k$ does the following system of equations possess a non-trivial solution over the

Question

For what value of $k$ does the following system of equations possess a non-trivial solution over the set of rationals? $x+k y+3 z=0,3 x+k y-2 z=0,2 x+3 y-4 z=0$

Professor PC Thomas Classes · Answer

For non-trivial solution, $D=0$\ or $\left|\begin{array}{ccc}1 & k & 3 \ 3 & k & -2 \ 2 & 3 & -4\end{array}ight|=0$\ Applying $R_{2} ightarrow R_{2}-3 R_{1}, R_{3} ightarrow R_{3}-2 R_{1}$,\ we get $\Delta=\left|\begin{array}{ccc}1 & k & 3 \ 0 & -2 k & -11 \ 0 & 3-2 k & -10\end{array}ight|=0$\ or $\quad 20 k+11(3-2 k)=0 \quad 	herefore \quad k=\frac{33}{2}$\ Putting the value of $k$ the equations are $$\begin{equation*} x+\frac{33}{2} y+3 z=0 	ag{i} \end{equation*}$$ $3 x+\frac{33 y}{2}-2 z=0 \Rightarrow 2 x+3 y-4 z=0$\ Eliminating $x$, we get $3 y=-z$\ Let $y=\lambda, z=-3 \lambda$ from (i) $x=\frac{-15}{2} \lambda$\ All solutions are $x=\frac{-15 \lambda}{2}, y=\lambda, z=-3 \lambda$ where $\lambda$ is rational.\