 Solution

The defining condition for an antisymmetric matrix forcesIts dot product with $(x,y{)}^T$ is zero, and its Euclidean norm is $|a|\sqrt{x^2+y^2}$. Thus it combines a right-angle rotation with a uniform dilation by $|a|$. More precisely, the rotation is clockwise for $a>0$ and anticlockwise for $a<0$. A direction-independent description is a clockwise right-angle rotation followed by multiplication by the signed scalar $a$.

A second application givesGeometrically this is a half-turn combined with a dilation by $a^2$. Because $A$ is nonzero, $a\ne 0$.