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

Answer: 0.5

If the system of equations has non-trivial solutions, then the determinant of coefficient matrix is zero
$\left|\begin{array}{ccc}1& -c& -c\\ c& -1& c\\ c& c& -1\end{array}\right|=0$
$\left(1-c^2\right)+c\left(-c-c^2\right)-c\left(c^2+c\right)=0$
$\left(1+c\right)\left(1-c\right)-2c^2\left(1+c\right)=0$
$\left(1+c\right)\left(1-c-2c^2\right)=0$
${\left(1+c\right)}^2\left(1-2c\right)=0$
$c=-1$ 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.