Question

The greatest value of $c\in R$ for which the system of linear equations
$x - cy - cz = 0$
$cx - y + cz = 0$
$cx + cy - z = 0$
has a non-trivial solution, is

Answer 1

If the system of equations has non-trivial solutions, then the determinant of coefficient matrix is zero
$\begin{vmatrix}1&-c&-c\\ c&-1&c\\ c&c&-1\end{vmatrix} =0$
$\left(1 -c^{2}\right) +c\left(-c -c^{2}\right) -c\left(c^{2} +c\right) =0$
$\left(1 +c\right)\left(1 -c\right) -2c^{2} \left(1+c\right) =0$
$\left(1 +c\right) \left(1 -c -2c^{2}\right) =0$
$\left(1 +c\right)^{2} \left(1 -2c\right) =0$
$c = -1$ or $\frac{1}{2}$
Hence, the greatest value of $c$ is $\frac{1}{2}$for which the system of linear equations has non-trivial solution.