$A^{2}=\left[\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right]\left[\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right]$

$=\left[\begin{array}{cc}1 & -a+a b \\ 0 & b^{2}\end{array}\right]$

$\therefore T _{ n }=\left\{ A \in S ; A ^{ n ( n +1)}= I \right\}$

$\therefore$ $b$ must be equal to $1$

$\therefore$ In this case $A ^{2}$ will become identity matrix and a can take any value from $1$ to $100$

$\therefore$ Total number of common element will be $100$ .