1. matrices question

Question: By considering the effect on the unit square or otherwise write down the matrices M and R which represent a reflection in the line y=x and a rotation about the origin through an agnle $\displaystyle \theta$, respectively, Find the matrix $\displaystyle M^{-1}RM$ and describe it in words. Find the matrix product $\displaystyle R^{-1}MR$ and show that $\displaystyle R^{-1}MR$ is

$\displaystyle \begin{pmatrix} sin 2\Theta & cos 2\Theta\\ cos 2\Theta & -sin 2\Theta \end{pmatrix}$

After doing this question I wanted to ask if

$\displaystyle R^{-1}MR =$$\displaystyle \begin{pmatrix} sin 2\Theta & cos 2\Theta\\ cos 2\Theta & -sin 2\Theta \end{pmatrix}$ is

generally the required matrix for a reflection in the line $\displaystyle y=-mx$

am i right in assuming that?

thanks for helping

2. Re: matrices question

No, you are not. For one thing, a "reflection" cannot be represented as a rotation. For another, there is no angle "$\displaystyle \theta$" associated with a reflection. The key point here is that a reflection in the line y= x "swaps" x and y, (x, y) becomes (y, x). In particular, (1, 0) is mapped to (0, 1) and (0, 1) is mapped to (1, 0). The matrix that does that is the matrix having (0, 1) and (1, 0) as columns, in that order: $\displaystyle R= \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix}y \\ x \end{pmatrix}$.

The matrix corresponding to "rotation through angle $\displaystyle \theta$" is $\displaystyle M= \begin{pmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}$. The inverse to "rotate through angle $\displaystyle \theta$" is "rotate through angle $\displaystyle -\theta$ so instead of inverting matrix M we can just write $\displaystyle M^{-1}= \begin{pmatrix}cos(-\theta) & -sin(-\theta) \\ sin(-\theta) & cos(-\theta)\end{pmatrix}= \begin{pmatrix}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{pmatrix}$

Now, calculate $\displaystyle M^{-1}RM$ using those matrices.