Property on glide reflections I need to prove that the conjugate of a glide reflection is a glide reflection.
What I have tried: Let $m: X= \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} \cos \phi & \sin \phi \\ \sin \phi & -\cos \phi \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}+\begin{pmatrix} a_1\\ a_2 \end{pmatrix}$ 
and $n:X= \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} \cos \psi & \sin \psi \\ \sin \psi & -\cos \psi \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}+\begin{pmatrix} b_1\\ b_2 \end{pmatrix}$  
I wanted to calculate $n \circ m \circ n^{-1}$ and show that it has the form of a glide reflection but what is $n^{-1}$ ?

Hint: The involved matrix, let's call her $M_{\psi}$, is invertible so simply solve $y=M_{\psi}x+b$ for $x$.
