[Math] Find the matrix of linear transformation (in Standard basis) that rotates clockwise every vector

Question

Find the matrix of linear transformation (in Standard basis) that rotates clockwise every vector in $ \mathbb{R}^{2}$ through an angle $ \pi/4 $ and then reflects it across y axis. $$ $$ The standard matrix of rotation by $ \pi/4 \ clockwise $ is R = $ \begin{pmatrix} cos(\pi/4) & \sin (\pi/4) \\ -\sin(\pi/4) & cos(\pi/4) \end{pmatrix} $= $ \begin{pmatrix} 1/\sqrt2 & 1/\sqrt 2 \\ -1/\sqrt 2 & 1/\sqrt 2 \end{pmatrix} $ . Now reflection matrix about y axis is
R'= $ \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ . Hence composition of Rotation and Reflection is $ \ R \circ R'=\begin{pmatrix} 1/\sqrt 2 & 1/\sqrt 2 \\ -1/\sqrt 2 & 1/\sqrt 2 \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $ . But I am not sure , please help me

Answer 1

The product of two matrices is not commutative. If you want the matrix that represents first the rotation than the reflection, the correct order is $R'\cdot R$.