JEE Main 2021 (Online) 3rd August Morning ShiftQuestion 1
Let $A=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}$ and $B=\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}$. Then the set $\{n\in\mathbf{N}\mid A^{n}+A^{n+1}+A^{n+2}=B\}$ is
Matrices
Solution
$A$ is a cyclic permutation matrix with $A^3=I$. For any $n$, $A^n+A^{n+1}+A^{n+2}=A^n(I+A+A^2)=I+A+A^2$ (since $\{A^n,A^{n+1},A^{n+2}\}=\{I,A,A^2\}$), which is the all-ones matrix $B$. So it holds for all $n$: the set is $\mathbf N$. Option (D).