JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Matrices

Direct multiplication gives

Multiplying any matrix of the form $M=\left(\begin{array}{ccc}1& 0& 0\\ m& 1& 1\\ 1& 0& 0\end{array}\right)$ by $A$ on the left reproduces the first and third rows and replaces the second row by the sum of the second and third rows of $M$, which raises $m$ by $1$. Hence

Therefore $A^{2025}-A^{2020}$ has all entries zero except the one in the second row and first column, which is $2024-2019=5$. The same matrix is $A^6-A$, whose corresponding entry is $5-0=5$, so $A^{2025}-A^{2020}=A^6-A$.