Let $\mathrm{A}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right). \mathrm{Then} {\mathrm{A}}^{2025}-{\mathrm{A}}^{2020}$ is equal to

$\begin{array}{rcl}\left|\mathrm{A}-\mathrm{\lambda I}\right|& =& \left|\begin{array}{ccc}1-\mathrm{\lambda }& 0& 0\\ 0& 1-\mathrm{\lambda }& 1\\ 1& 0& -\mathrm{\lambda }\end{array}\right|=0\\ {\mathrm{\lambda }}^3-2{\mathrm{\lambda }}^2+\mathrm{\lambda }& =& 0\\ {\mathrm{A}}^3-2{\mathrm{A}}^2+\mathrm{A}& =& 0\\ {\mathrm{A}}^2-\mathrm{A}& =& {\mathrm{A}}^3-{\mathrm{A}}^2={\mathrm{A}}^4-{\mathrm{A}}^3={\mathrm{A}}^5-{\mathrm{A}}^4={\mathrm{A}}^6-{\mathrm{A}}^5={\mathrm{A}}^7-{\mathrm{A}}^6\\ \therefore {\mathrm{A}}^7-{\mathrm{A}}^2& =& {\mathrm{A}}^6-\mathrm{A}\\ & \Rightarrow & {\mathrm{A}}^8-{\mathrm{A}}^3={\mathrm{A}}^7-{\mathrm{A}}^2={\mathrm{A}}^6-\mathrm{A}\\ & & \mathrm{and} \mathrm{so} \mathrm{on},\\ {\mathrm{A}}^{2025}-{\mathrm{A}}^{2020}& =& {\mathrm{A}}^6-\mathrm{A}\end{array}$