I'm really stuck trying to get the key in this cryptosystem. I've tried it numerous times, each time getting different (and wrong!) answers for matrices A and B. If you could help that would be great! A correct answer would be great then I can fill in the steps and see where I went wrong.

The plaintext and is written in the alphabet a,b,...,z,ä. And cipher text A,B,...Z,Ä.

So the alphabets both have 27 letters.

Consider an affine matrix cipher on diagraphs and we have a crib: needle is encrypted as TPFDFH. Determine the fomulae for encryption and decryption.

Basically you have to solve these equations for the 2x2 matrix A and the 2x1 matrix B.