Let A=1 0 00 1 11 0 0 then A2025−A2020=

A=100010100⇒A2=100111100 ⇒A3=100211100, ⇒A4=100311100 ⇒An=100n−111100∴A2025−A2020=000500000=A6−A

Let A=1 0 00 1 11 0 0 then A2025−A2020=

A=100010100⇒A2=100111100 ⇒A3=100211100, ⇒A4=100311100 ⇒An=100n−111100∴A2025−A2020=000500000=A6−A

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$Let A = (\begin{array}{lll} 1 0 0 \\ 0 1 1 \\ 1 0 0 \end{array}) then A^{2025} - A^{2020} =$

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$A^{5}$

$A^{6} - A$

A⁶

$A^{5} - A$

answer is B.

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Detailed Solution

$\begin{array}{l} A = [\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}] \Rightarrow A^{2} = [\begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \end{array}] \Rightarrow A^{3} = [\begin{array}{lll} 1 & 0 & 0 \\ 2 & 1 & 1 \\ 1 & 0 & 0 \end{array}], \Rightarrow A^{4} = [\begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & 1 \\ 1 & 0 & 0 \end{array}] \Rightarrow A^{n} = [\begin{array}{ccc} 1 & 0 & 0 \\ n - 1 & 1 & 1 \\ 1 & 0 & 0 \end{array}] \\ ∴ A^{2025} - A^{2020} = [\begin{array}{lll} 0 & 0 & 0 \\ 5 & 0 & 0 \\ 0 & 0 & 0 \end{array}] = A^{6} - A \end{array}$