Area of an ellipse. 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?

It is a known formula that the area enclosed in the ellipse with semi-axes $a$ and $b$ is $\pi ab$, as may be seen from the orthogonal affinity that transforms a circle into an ellipse.
