The greatest value of c ∈ R for which the system of linear equations x - c y - c z = 0 , c x - y + c z…

$\because$ System of equation has non trivial solution. It is possible only when $∆=0$.
$\therefore ∆ =\left|\begin{array}{ccc}1& -c& -c\\ c& -1& c\\ c& c& -1\end{array}\right|=0$
$\Rightarrow 1\left(1-c^2\right)+c\left(-c-c^2\right)-c\left(c^2+c\right)=0$
$\Rightarrow {2c}^3+3c^2-1=0$
$\Rightarrow c=-1, \frac{1}{2}$

Hence, the greatest value of $c=\frac{1}{2}$.