Question

Let A be a square matrix of order two such that $A-A{A}^T=I_2$. If trace $\left(A\right)\ne R$ and entries are complex numbers, number of such matrices possible are

Answer 1

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]-\left[\begin{array}{cc}{a}^2+b^2& ac+bd\\ ac+bd& c^2+d^2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$ac+bd=b=c\Rightarrow (a+d)b=0\Rightarrow b=0$
$\Rightarrow a^2-a+1=0,d^2-d+1=0$
$a=-w,-w^2,d=-w,-w^2$
So two matrices possible.
$\left[\begin{array}{cc}-w& 0\\ 0& -w\end{array}\right],\left[\begin{array}{cc}-w^2& 0\\ 0& -w^2\end{array}\right]$