Let S be the set of  $2\times 2$ matrices given by $S=\{A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],wherea,b,c,d\in I\}$, such that $A^T=A^{-1}$. Then which of the following is/are correct?

As  $A{A}^T=I_2$

$$\begin{gathered} \Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow a=0,b=\pm 1,d=0,c=\pm 1 \end{gathered}$$

Total 8 matrices are possible
They are
$$\begin{gathered} \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right] \\ Also, |A-I_2|=|A-A{A}^T|=\left|A\right||I_2-A^T| \\ =\left|A\right|\left|{(I_2-A^T)}^T\right|=\left|A\right||I_2-A| \\ =\left|A\right||A-I_2| \\ \Rightarrow \left|A\right|=1(As,|A-I_2|\ne 0) \\ \text{except\hspace{0.17em}\hspace{0.17em}}A=I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ Where \left|A\right|=1 but \\ |A-I_2|=0 \end{gathered}$$