JEE Main 2021 (Online) 26th August Evening ShiftQuestion 1
Let $A=\begin{pmatrix}1&0&0\\0&1&1\\1&0&0\end{pmatrix}$. Then $A^{2025}-A^{2020}$ is equal to :
Matrices
Solution
Rows: $A^n$ has row 1 $=(1,0,0)$, row 3 $=(1,0,0)$ and row 2 $=(n-1,1,1)$ for $n\ge2$ (induction: row 2 of $A^{n+1}$ = row 2 + row 3 of $A^n$ = $(n,1,1)$). So $A^{2025}-A^{2020}$ has only entry $5$ at position $(2,1)$. $A^6-A$: row 2 is $(5,1,1)-(0,1,1)=(5,0,0)$, other rows $0$. Equal. Option (C).