The greatest value of c ∈ R for which the system of linear...

$∵$ System of equation has non trivial solution. It is possible only when $∆ = 0$.
∴ ∆ = \left|\begin{cases}1 & - c & - c \\ c & - 1 & c \\ c & c & - 1\end{cases}\right| = 0
$⇒ 1\left(1 - {c}^{2}\right) + c\left(- c - {c}^{2}\right) - c\left({c}^{2} + c\right) = 0$
$⇒ {2c}^{3} + 3{c}^{2} - 1 = 0$
$⇒ c = - 1, \dfrac{1}{2}$

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