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

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.
$∴$ $\bigcap_{n=1}^{100}$ $T_n = 100$