JEE MAIN - Mathematics (2021 - 25th July Morning Shift - No. 22)
Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{
0 & i \cr
1 & 0 \cr
} } \right)}^n}\left( {\matrix{
a & b \cr
c & d \cr
} } \right) = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)\forall a,b,c,d \in R} \right.} \right\}$$, where i = $$\sqrt { - 1} $$. Then the number of 2-digit numbers in the set S is _____________.
Answer
11
Explanation
Let $$X = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)$$ & $$A = {\left( {\matrix{
0 & i \cr
1 & 0 \cr
} } \right)^n}$$
$$\Rightarrow$$ AX = IX
$$\Rightarrow$$ A = I
$$ \Rightarrow {\left( {\matrix{ 0 & i \cr 1 & 0 \cr } } \right)^n} = I$$
$$ \Rightarrow {A^8} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$
$$\Rightarrow$$ n is multiple of 8
So, number of 2 digit numbers in the set
S = 11 (16, 24, 32, .........., 96)
$$\Rightarrow$$ AX = IX
$$\Rightarrow$$ A = I
$$ \Rightarrow {\left( {\matrix{ 0 & i \cr 1 & 0 \cr } } \right)^n} = I$$
$$ \Rightarrow {A^8} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$
$$\Rightarrow$$ n is multiple of 8
So, number of 2 digit numbers in the set
S = 11 (16, 24, 32, .........., 96)
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