Question

Let $A= \begin{bmatrix}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{bmatrix}$ and $B=A$ - I. If $\omega=\frac{\sqrt{3} i-1}{2}$, then the number of elements in the set $\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }+(\omega B )^{ n }\right.$ $= A + B \}$ is equal to ____

Answer 1

$ A= \begin{bmatrix} 2 & -1 & -1 \\1 & 0 & -1 \\ 1 & -1 & 0\end{bmatrix} \Rightarrow A^2=A \Rightarrow A^n=A $
$ \forall n \in\{1,2, \ldots, 100\} $
Now, $ B = A - I =\begin{bmatrix} 1 & -1 & -1 \\1 & -1 & -1 \\1 & -1 & -1\end{bmatrix} $
$B ^2 =- B $
$\Rightarrow B ^3 =- B ^2= B$
$ \Rightarrow B ^5= B$
$\Rightarrow B ^{99}= B$
Also, $\omega^{3 k }=1$
So, $n =$ common of $\{1,3,5, \ldots, 99\}$ and $\{3,6,9, \ldots, 99\}=17$