Question

The values of $\lambda$ for which the system of equations
$(\lambda +5)x+(\lambda -4)y+z=0$
$(\lambda -2)x+(\lambda +3)y+z=0$
$\lambda x+\lambda y+z=0$
has a non-trivial solution is (are)

• A. $-1,2$
• B. $0,-1$
• C. $0$
• D. none of these

Answer 1

The correct option is D none of these

Given:

$(\lambda +5)x+(\lambda -4)y+z=0$
$(\lambda -2)x+(\lambda +3)y+z=0$
$\lambda x+\lambda y+z=0$

Let $\Delta =\left|\begin{array}{ccc}\lambda +5& \lambda -4& 1\\ \lambda -2& \lambda +3& 1\\ \lambda & \lambda & 1\end{array}\right|$

Applying $R_1\to R_1-R_3,R_2\to R_2-R_3$

$\Delta =\left|\begin{array}{ccc}5& -4& 0\\ -2& 3& 0\\ \lambda & \lambda & 1\end{array}\right|$

Expanding along $C_3$

$\Delta =15-8=7\ne 0$