AES decryption of MixColumn

In the encryption of MixColumn you use the constants 01, 02 and 03. I don't understand why the inverse of 01 becomes both 0D and 09. I also don't understand how you get 01, 02 and 03 in the first place which is probably why the decryption doesn't make sense to me. I would appreciate any help with understanding how to calculate these values.

• Note that the elements of the inverse matrix are not the inverses of the individual elements; hence 0D and 09 are not "inverses" of 01 – poncho Dec 10 '18 at 19:35
• and, you calculate the inverse of the matrix in $GF(2^8)$ see How to invert matrix in finite field – kelalaka Dec 10 '18 at 19:36

The MixColumns $$MC$$ and InvMixColumns $$IMC$$ implement matrix multiplications. These two matrices defined over Galois Field $$GF(2^8)$$ and they are inverse of each other.
$$MC \times IMC = I_4$$
To find the inverse, one can use the Gaussian Elimination method. But finding the inverse a bit tricky, since you will need the inverse of the pivot elements. The inverse in $$GF(2^8)$$ can be found by using the Extended Euclidean Algorithm.