Consider the linear transformation, $\displaystyle T:\mathbb{R}^2\rightarrow \mathbb{R}^2$, for which $\displaystyle T(x)$ is found by first rotating the point $\displaystyle x$ through an angle of $\displaystyle \frac{\pi}{3}$ anticlockwise about the origin and then reflecting that point across the line $\displaystyle y = x$.

Find the 2x2 matrix A for which $\displaystyle T(x) = Ax$

I understand how to transform matrices in the required ways described above, what I don't understand is which matrix I am transforming to begin with, i.e., to rotate a matrix anticlockwise bt \frac{\pi}{3} I just multiply it by:

$\displaystyle \left(\begin{matrix} cos(\frac{\pi}{3}) &-sin(\frac{\pi}{3}) \\sin(\frac{\pi}{3}) & cos(\frac{\pi}{3})\end{matrix}\right)$

Similarly, to reflect a matrix across the line y = x I just multiply it by:

$\displaystyle \left(\begin{matrix} 0 &1\\1 & 0\end{matrix}\right)$

What I don't understand in this question is which matrix I am transforming in the first place? What does rotating the point x mean? What matrix am I transforming by multiplying it by:

$\displaystyle \left(\begin{matrix} cos(\frac{\pi}{3}) &-sin(\frac{\pi}{3}) \\sin(\frac{\pi}{3}) & cos(\frac{\pi}{3})\end{matrix}\right)$$\displaystyle \left(\begin{matrix} 0 &1\\1 & 0\end{matrix}\right) = $$\displaystyle \left(\begin{matrix} -sin(\frac{\pi}{3}) &cos(\frac{\pi}{3}) \\cos(\frac{\pi}{3}) & sin(\frac{\pi}{3})\end{matrix}\right)$

Which matrix is A and which is x? do I need to multiply everything by the column vector $\displaystyle (x_1,x_2)$ or something?

Thanks.