Using the encryption matrix from AES, how do you compute the decryption matrix?
First of all, this is not "AES", but something else.  Someone might claim that it's somewhat similar to AES, but it's still not AES.

However, to answer your question:


  please let me if I am on the right track or I'm not even close to the start of it all


Actually, you already got your answer:


  Compute the decryption matrix $D = E^{-1}$


That is, compute the matrix $D$ such that $D$ and $E$, multiplied together using standard matrix multiplication, gives:

$$\begin{bmatrix}d_{00}&amp;d_{01}\\d_{10}&amp;d_{11}\end{bmatrix}\begin{bmatrix}1&amp;x^2 \\x^2&amp;1\end{bmatrix}= \begin{bmatrix}1&amp;0\\0&amp;1\end{bmatrix}$$

This is a standard matrix inversion problem, and all the standard tools used to invert matrices over $\mathbb{R}$ work (and as a bonus, you don't have to worry about numeric stability).  Now, you do have to do your internal computations in $\mathbb{F}_{16}$, but that's pretty easy.