Question

If $X$ is a non-zero column matrix, such that $A X=\lambda X$ where $\lambda$ is a scalar and the matrix $A$ is $\begin{bmatrix}4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -5 & -2\end{bmatrix}$, then sum of distinct values of $\lambda$ is

Answer 1

$A X=\lambda X \Rightarrow(A-\lambda I) X=0$
but $X \neq 0 \Rightarrow|A-\lambda I|=0 $
$\Rightarrow \begin{vmatrix}4-\lambda & 6 & 6 \\1 & 3-\lambda & 2 \\-1 & -5 & -2-\lambda\end{vmatrix}=0 $
$\Rightarrow(\lambda-1)(\lambda-2)^{2}=0$
$ \Rightarrow \lambda=1,2,2$
Sum of distinct values $1+2=3$