JEE MAIN - Mathematics (2021 - 26th August Evening Shift - No. 2)
Let $$A = \left( {\matrix{
1 & 0 & 0 \cr
0 & 1 & 1 \cr
1 & 0 & 0 \cr
} } \right)$$. Then A2025 $$-$$ A2020 is equal to :
A6 $$-$$ A
A5
A5 $$-$$ A
A6
Explanation
$$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 1 & 1 \cr
1 & 0 & 0 \cr
} } \right] \Rightarrow {A^2} = \left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 1 \cr
1 & 0 & 0 \cr
} } \right]$$
$${A^3} = \left[ {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right] \Rightarrow {A^4} = \left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right]$$
$${A^n} = \left[ {\matrix{ 1 & 0 & 0 \cr {n - 1} & 1 & 1 \cr 1 & 0 & 0 \cr } } \right]$$
$${A^{2025}} - {A^{2020}} = \left[ {\matrix{ 0 & 0 & 0 \cr 5 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right]$$
$${A^6} - A = \left[ {\matrix{ 0 & 0 & 0 \cr 5 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right]$$
$${A^3} = \left[ {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right] \Rightarrow {A^4} = \left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right]$$
$${A^n} = \left[ {\matrix{ 1 & 0 & 0 \cr {n - 1} & 1 & 1 \cr 1 & 0 & 0 \cr } } \right]$$
$${A^{2025}} - {A^{2020}} = \left[ {\matrix{ 0 & 0 & 0 \cr 5 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right]$$
$${A^6} - A = \left[ {\matrix{ 0 & 0 & 0 \cr 5 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right]$$
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