Let $\alpha$ be the endomorphism given by
$\alpha$:$\mbox{} \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]$$\rightarrow$ $\mbox{} \left[ \begin{array}{cc} d & -b \\ -c & a \end{array} \right]$
I need to find the eigenvalues and associate eigenspace. Since the determinants are equal, does that mean that the product of the eigenvalues is also the same? And which matrix ought I use to find the eigenvalues?