Question

The value of $C$ for which the system of equations have non trivial solution is:
$cx-y-z=0$
$-cx+y-cz=0$
$x+y-cz=0$

• A. $\frac{1}{2}$
• B. $-1$
• C. $2$
• D. $0$

Answer 1

The correct option is B $-1$

For the given system of equations to have non trival solution :
$\Delta =0$
$\Rightarrow \Delta =\left|\begin{array}{ccc}C& -1& -1\\ -C& 1& -C\\ 1& 1& -C\end{array}\right|=0$
$R_1\to R_1+R_2$
$\Rightarrow \left|\begin{array}{ccc}0& 0& -(1+C)\\ -C& 1& -C\\ 1& 1& -C\end{array}\right|=0$
$\Rightarrow -(1+C)(-C-1)=0$
$\Rightarrow (1+C{)}^2=0$
$\Rightarrow C=-1$