I need to find the area of the image of a circle centred at the origin with radius 3 under the transformation:

$ \begin{pmatrix} 3 & 0\\ 0 & \frac{1}{3} \end{pmatrix} $

The image is the ellipse $ \frac{x^2}{81}+y^2=1$. It would appear that it has the same area as the original circle i.e. $9\pi$. Is this because the matrix has some special property such as being its own inverse?