Answer:
Upon multiplication with this matrix A, it gives a vector perpendicular to the original, with a magnitude of [tex]\sqrt{Det|A|}[/tex] times of the original.
If a 2x2 matrix A has complex eigenvalues, this is an antisymmetric matrix.
Therefore:
[tex]A=\left[\begin{array}{cc}0&a\\-a&0\end{array}\right] \\Det|A|=a^2[/tex]
if we multiply (x,y) to this matrix A:
[tex]A\cdot (x,y)^T=\left[\begin{array}{cc}0&a\\-a&0\end{array}\right](x,y)^T=(ay,-ax)^T=\sqrt{Det|A|} (y,-x)^T[/tex]
