So... each letter from A to Z corresponds to a number 0 to 25. Now I want to find the pairs (x,y) such that in $\displaystyle \mathbb{Z}_{26}$,

$\displaystyle \begin{bmatrix} 8 & 3\\ 1 & 7 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} x\\ y \end{bmatrix}$

And we are only told that for (A, A), which corresponds to (0,0), this holds.

So, how do I find all pairs x, y such that they are unchanged by the multipication? Going through all possible combinations of x and y is a real daunting task. Is there an easy way for finding all the pairs?

P.S. In $\displaystyle \mathbb{Z}_{26}$,

$\displaystyle K^{-1} = \begin{bmatrix} 7 & 23\\ 25 & 8 \end{bmatrix}$.