JEE MAIN - Mathematics (2022 - 28th June Evening Shift - No. 26)
Let $$A = \left( {\matrix{
{1 + i} & 1 \cr
{ - i} & 0 \cr
} } \right)$$ where $$i = \sqrt { - 1} $$. Then, the number of elements in the set { n $$\in$$ {1, 2, ......, 100} : An = A } is ____________.
Answer
25
Explanation
$$\therefore$$ $${A^2} = \left[ {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right]\left[ {\matrix{ {1 + i} & 1 \cr { - 1} & 0 \cr } } \right] = \left[ {\matrix{ i & {1 + i} \cr {1 - i} & { - i} \cr } } \right]$$
$${A^4} = \left[ {\matrix{ i & {1 + i} \cr {1 - i} & { - i} \cr } } \right]\left[ {\matrix{ i & {1 + i} \cr {1 - i} & { - i} \cr } } \right] = I$$
So A5 = A, A9 = A and so on.
Clearly n = 1, 5, 9, ......, 97
Number of values of n = 25
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