Question

The number of distinct real values of $\lambda$, for which the determinant $\left|\begin{array}{ccc}-{\lambda }^2& 1& 1\\ 1& -{\lambda }^2& 1\\ 1& 1& -{\lambda }^2\end{array}\right|$ vanishes, is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

$R_1\to R_1+R_2+R_3$
$\left(2-{\lambda }^2\right)\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\lambda }^2& 1\\ 1& 1& -{\lambda }^2\end{array}\right|=0$
$C_1\to C_1-C_2$ and $C_2\to C_2-C_3$
$\left(2-{\lambda }^2\right)\left|\begin{array}{ccc}0& 0& 1\\ 1+{\lambda }^2& -{\lambda }^2-1& 1\\ 0& 1+{\lambda }^2& -{\lambda }^2\end{array}\right|=0\Rightarrow \left(2-{\lambda }^2\right){\left[1+{\lambda }^2\right]}^2=0$
$\therefore {\lambda }^2=2\Rightarrow \lambda =\pm \sqrt{2}\Rightarrow$ two values of $\lambda$