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  4. Let A= beginpmatrix1 0 0 0 1 1 1 0 0 endpmatrix . Then A2025-A2020 is equal to

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Q. Let $A=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{pmatrix} .$ Then $A^{2025}-A^{2020}$ is equal to

JEE MainJEE Main 2021Matrices

Solution:

$A =\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{bmatrix} \Rightarrow A ^{2}=\begin{bmatrix}1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 0\end{bmatrix}$
$A^{3}=\begin{bmatrix}1 & 0 & 0 \\ 2 & 1 & 1 \\ 1 & 0 & 0\end{bmatrix}\Rightarrow A^{4}=\begin{bmatrix}1 & 0 & 0 \\ 3 & 1 & 1 \\ 1 & 0 & 0\end{bmatrix}$
$A^{n}=\begin{bmatrix}1 & 0 & 0 \\ n-1 & 1 & 1 \\ 1 & 0 & 0\end{bmatrix}$
$A^{2025}-A^{2020}=\begin{bmatrix}0 & 0 & 0 \\ 5 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$
$A^{6}-A=\begin{bmatrix}0 & 0 & 0 \\ 5 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$