\[x'=Ax+a\]
and \[yBy+b\]
are two affine transformation. If the second transformation is the inverse of the first, can we find the second trandforemation in terms of the first? Yes we can.\[x=Bx'+b\]
and substitute \[x'=Ax+a\]
to give\[x=B(Ax+a)+b\]
\[x=BAx+Ba+b\]
\[0=BAx-x+Ba+b\]
\[0=(BA-I)+Ba+b\]
Then from the first term on the right
\[B=A^{-1
\]
and from the second and third terms \[0=Ba+b \rightarrow b=-Ba=aA^{-1}a\]