Question

Number of real values of $\lambda$ for which the system of equations
$(\lambda +3)x+(\lambda +3)y+z=0$
$3x+(\lambda +3)y+z=0$
$2x+3y+z=0$
has a non-trivial solutions is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is B $1$

Let $\Delta =\left|\begin{array}{ccc}\lambda +3& \lambda +3& 1\\ 3& \lambda +3& 1\\ 2& 3& 1\end{array}\right|=0$
Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$
$\Delta =\left|\begin{array}{ccc}\lambda +3& \lambda +3& 1\\ -\lambda & 0& 0\\ -1-\lambda & -\lambda & 0\end{array}\right|=0\Rightarrow (-\lambda )(-\lambda )-0(-1-\lambda )=0\Rightarrow {\lambda }^2=0\Rightarrow \lambda =0$