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

$\mathrm{A}=\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\\ 1& -1& 0\end{array}\right]\Rightarrow {\mathrm{A}}^2=\mathrm{A}\Rightarrow {\mathrm{A}}^{\mathrm{n}}=\mathrm{A}$.               $\mathrm{\forall }\mathrm{n}\in \{1,2,\dots ,100\}$

Now, $\mathrm{B}=\mathrm{A}-\mathrm{I}=\left[\begin{array}{r}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\end{array}\right]$

$\begin{array}{l}{\mathrm{B}}^2=-\mathrm{B}\\ \Rightarrow {\mathrm{B}}^3=-{\mathrm{B}}^2=\mathrm{B}\\ \Rightarrow {\mathrm{B}}^5=\mathrm{B}\\ \Rightarrow {\mathrm{B}}^{99}=\mathrm{B}\end{array}$

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

So, n = number of elements in  the intersection of   {1, 3,  5, …, 99} and {3, 6, 9, …, 99} 

= 17