Question

Let \(S ={  (\begin{matrix}   -1 & 0 \\   a & b \end{matrix}), a,b, ∈(1,2,3,.....100)}\) and 
let \(T_n = {A ∈ S : A^{n(n + 1)} = I}. \)
Then the number of elements in \(\bigcap_{n=1}^{100}\) \(T_n \) is

Answer 1

Correct Answer: 100

The correct answer is 100
\(S ={  (\begin{matrix}   -1 & 0 \\   a & b \end{matrix}), a,b, ∈(1,2,3,.....100)}\)
\(∴ A =\) \((\begin{matrix}   -1 & 0 \\   a & b \end{matrix})\)
then even powers of A as
\(A(\begin{matrix}   1 & 0 \\   0 & 1 \end{matrix})\)
if b = 1 and a ∈ {1,….., 100}
Here, n(n + 1) is always even.
∴ \(T_1, T_2, T_3\), …, \(T_n\) are all I for b = 1 and each value of a.
\(∴\) \(\bigcap_{n=1}^{100}\) \(T_n = 100\)