Find the standard matrix for the composition of the following two linear operators on $\Bbb R^2$: A reflection about the line $y = x$, followed by a rotation counterclockwise of $60^\circ$.
I'm not sure if it's the standard matrix of reflection about y=x multiply by the standard matrix of rotation,and then plug in θ=60°.
This is my work so far:
$$ T(x, y) = (y, x)\\ T(\vec e_1) = T(1, 0) = (0, 1)\\ T(\vec e_2) = T(0, 1) = (1, 0) $$so the standard matrix for the reflection transformation is: $\left[\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right]$
The standard matrix for rotation transformation: $\left[\begin{smallmatrix}\cos \theta & \,-\sin\theta\\\sin\theta & \cos\theta\end{smallmatrix}\right]$ $$ \left[\begin{matrix}0&1\\1&0\end{matrix}\right]\left[\begin{matrix}\cos \theta & \,-\sin\theta\\\sin\theta & \cos\theta\end{matrix}\right] = \left[\begin{matrix}\sin \theta & \cos\theta\\\cos\theta & \,-\sin\theta\end{matrix}\right] $$ Let the matrix $A$ be the standard matrix after two transformations $$ A = \left[\begin{matrix}\cos 60^\circ & \,-\sin60^\circ\\\sin60^\circ & \cos60^\circ\end{matrix}\right] = \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt3}{2}\\\frac{\sqrt{3}}{2} & \frac12\end{bmatrix} $$