Mathematics - Matrices Question with Solution | TestHub

$A=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right]$

$A^2=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 1\\ 1& 0& 0\end{array}\right]$$A^3=\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 1\\ 1& 0& 0\end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 2& 1& 1\\ 1& 0& 0\end{array}\right]$

$A^4=\left[\begin{array}{lll}1& 0& 0\\ 3& 1& 1\\ 1& 0& 0\end{array}\right]$

$A^n=\left[\begin{array}{ccc}1& 0& 0\\ n-1& 1& 1\\ 1& 0& 0\end{array}\right]$

$A^{2025}-A^{2020}=\left[\begin{array}{ccc}1& 0& 0\\ 2024& 1& 1\\ 1& 0& 0\end{array}\right]-\left[\begin{array}{ccc}1& 0& 0\\ 2019& 1& 1\\ 1& 0& 0\end{array}\right]$$=\left[\begin{array}{lll}0& 0& 0\\ 5& 0& 0\\ 0& 0& 0\end{array}\right]=A^6-A$