Question

Let $A=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$ and $X\ne 0$ be a column matrix such that $AX=\lambda X$, find sum of distinct values of $\lambda$

Answer 1

Given A =$$\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$$
Also given $AX=\lambda X$
$\Rightarrow (A-\lambda I)X=O$
$\Rightarrow A-\lambda I=0$ ($\because X\ne O$)
$\Rightarrow |A-\lambda I|=0$
$$\left|\begin{array}{ccc}-\lambda & 1& 1\\ 1& -\lambda & 1\\ 1& 1& -\lambda \end{array}\right|$$=0
$\Rightarrow (\lambda +1)(-{\lambda }^2+\lambda +2)=0$
$\Rightarrow \lambda =2,-1$
So, required sum $=2-1=1$