If P is an orthogonal matrix, $\mathrm{A}=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right],\mathrm{Q}={\mathrm{PAP}}^{\mathrm{T}}$ and ${\mathrm{P}}^{\mathrm{T}}{\mathrm{Q}}^{2023}\mathrm{P}=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{n}\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]$ then $n=$ _____

$$\begin{gathered} \begin{array}{l}{\mathrm{Q}}^2=\left({\mathrm{PAP}}^{\mathrm{T}}\right)\left({\mathrm{PAP}}^{\mathrm{T}}\right)\\ ={\mathrm{PA}}^2{\mathrm{P}}^{\mathrm{T}}\left({\mathrm{PP}}^{\mathrm{T}}={\mathrm{P}}^{\mathrm{T}}\mathrm{P}=\mathrm{I}\right)\\ {\mathrm{Q}}^{\mathrm{n}}={\mathrm{PA}}^{\mathrm{n}}{\mathrm{P}}^{\mathrm{T}}\Rightarrow {\mathrm{P}}^{\mathrm{T}}{\mathrm{Q}}^{\mathrm{n}}\mathrm{P}={\mathrm{A}}^{\mathrm{n}}\\ {\mathrm{A}}^2=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{ll}1& 2\\ 0& 1\end{array}\right]\\ {\mathrm{A}}^3=\left[\begin{array}{ll}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{ll}1& 3\\ 0& 1\end{array}\right]\end{array} \\ {A}^{2023}=\left[\begin{array}{cc}1& 2023\\ 0& 1\end{array}\right];\hspace{0.17em} \end{gathered}$$