Let A = [ cases 1 & 0 & 0 0 & 1 & 1 1 & 0 & 0 cases ] . Then A ^ 2025 - A ^ 2020 is equal to

Correct answer: A. A ^ 6 - A A = [ cases 1 & 0 & 0 0 & 1 & 1 1 & 0 & 0 cases ] A^ 2 = [ cases 1 & 0 & 0 0 & 1 & 1 1 & 0 & 0 cases ][ cases 1 & 0 & 0 0 & 1 & 1 1 & 0 & 0 cases ] = [ cases 1 & 0 & 0 1 & 1 & 1 1 & 0 & 0 cases ]A^ 3 = [ cases 1 & 0 & 0 1 & 1 & 1 1 & 0 & 0 cases ][ cases 1 & 0 & 0 0 & 1 & 1 1 & 0 & 0 cases ] = [ cases 1 & 0 & 0 2 & 1 & 1 1 & 0 & 0 cases ] A^ 4 = [ cases 1 & 0 & 0 3 & 1 & 1 1 & 0 & 0 cases ] A^ n = [ cases 1 & 0 & 0 n - 1 & 1 & 1 1 & 0 & 0 cases ] A^ 2025 - A^ 2020 = [ cases 1 & 0 & 0 2024 & 1 & 1 1 & 0 & 0 cases ] - [ cases 1 & 0 & 0 2019 & 1 & 1 1 & 0 & 0 cases ]= [ cases 0 & 0 & 0 5 & 0 & 0 0 & 0 & 0 cases ] = A^ 6 - A

JEE Main 202126 AugustEvening Shift

Exam: JEE Main
Year: 2021
Date: 26 August
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2021 · 26 August · Evening Shift · Mathematics · Matrices

Let A = [ cases 1 & 0 & 0 0 & 1 & 1 1 & 0 & 0 cases ] . Then A ^...

Let

A = \left[

\begin{cases}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{cases}

\right] .

Then

{A}^{2025} - {A}^{2020}

is equal to

Options

A ${A}^{6} - A$
B ${A}^{6}$
C ${A}^{5}$
D ${A}^{5} - A$

Correct answer

A. ${A}^{6} - A$

Step-by-step solution

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

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

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

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

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

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