The lines $p : y = 1$ and $q : x = −2$ are given in the affine plane $\mathbb{R}^2$. Find all affines mappings $A$ with the property $A(p) = q$ and $A(q) = p$. Hint: Where is such an affine mapping map $p∩q$?
Attempt: I know that the intersection of the lines is the point $(-2,1)$. I am assuming that $A(-2,1)=(-2,1)$, but how to justify this (if at all)? Where to use hint?