Inner Space Projection Using Matrices In my math class today, we proved that the ratio of the area of an inner space to that of the inner space projected by some matrix $A$ is equal to $|det(A)|$. In other words, if the area of an inner space is $a$, the area of that inner space projected by a matrix $M$ $= |det(M)|*a$ 
So, if I am given  the equation of a circle $x^2+y^2=1$, and the inner space of that circle is projected  by a matrix $M = \begin{pmatrix} 
1 & 2 \\
3 & 4 
\end{pmatrix}$, then the area of the ellipse would be $\pi *|det(M)|$, which equals $2\pi$
My question is, is there anyway to find said matrix given the two geometric shapes such that the projection of one shape by a matrix results in the other.
For example, if I am given a circle and an ellipse, and I know that the ratio of the areas is $R$, I understand that the determinant of the matrix would be $\pm R$, but is there a formula or method to compute the exact matrix?
Thanks in Advance!
P.S. If any of the terms I have used are incorrect, please let me know. I am new to MSE, as well as linear algebra, and any help is greatly appreciated!

If the transformation is linear then the matrix $M$ is determined if you know the images of two general points (i.e. two points that do not lie on the same line through the origin). Essentially, the entries in each row of $M$ are the solution to a pair of simultaneous linear equations.
Of course, in many cases the transformation of one shape into another will not be linear e.g. there is no linear transformation that maps a circle into a square.
