Let A = [ cases 2 & - 1 & - 1 1 & 0 & - 1 1 & - 1 & 0 cases ] and B...

Given, A = \left[\begin{cases}2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0\end{cases}\right]

⇒ {A}^{2} = \left[\begin{cases}2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0\end{cases}\right]\left[\begin{cases}2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0\end{cases}\right] = \left[\begin{cases}2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0\end{cases}\right] = A

$⇒ {A}^{n} = A$

Now $\forall n \in \left\{1, 2, \ldots , 100\right\}$

Now, B = A - I = \left[\begin{cases}1 & - 1 & - 1 \\ 1 & - 1 & - 1 \\ 1 & - 1 & - 1\end{cases}\right]

{B}^{2} = \left[\begin{cases}1 & - 1 & - 1 \\ 1 & - 1 & - 1 \\ 1 & - 1 & - 1\end{cases}\right]\left[\begin{cases}1 & - 1 & - 1 \\ 1 & - 1 & - 1 \\ 1 & - 1 & - 1\end{cases}\right] = - \left[\begin{cases}1 & - 1 & - 1 \\ 1 & - 1 & - 1 \\ 1 & - 1 & - 1\end{cases}\right] = - B

$⇒ {B}^{3} = - {B}^{2} = B$

$⇒ {B}^{5} = B$

$⇒ {B}^{99} = B$

Also, ${\omega}^{3k} = 1$

So, $n =$ common of \left\{1, 3, 5, \ldots , 99\right\}$and \left\{3, 6, 9, \ldots , 99\right\} = 17$