Matrices: Invariant lines / Lines of invariant points question

[Vantage]

I was working through a question page on matrix invariance, however an answer I got did not match the answer in the book. The question was as follows:

"R is a reflection through the line [imath]y = 2x[/imath]. S is a [imath]90^o[/imath] rotation anticlockwise about the origin. T is a stretch, scale factor 3, parallel to the y-axis."

From this, I gathered that in matrix forms: R = [imath]\left[ \begin{matrix} -0.6 & 0.8\\ 0.8 & 0.6 \end{matrix} \right][/imath], S = [imath]\left[ \begin{matrix} 0 & -1\\ 1 & 0 \end{matrix} \right][/imath] and T = [imath]\left[ \begin{matrix} 1 & 0\\ 0 & 3 \end{matrix} \right][/imath].

The question continues: "In each case, find any lines of invariant points and any other invariant lines through the origin."
The case I did not match the answer for was: [imath]S^{-1}RS[/imath].

(I worked out that [imath]S^{-1} = \left[ \begin{matrix} 0 & 1\\ -1 & 0 \end{matrix} \right][/imath] and that [imath]S^{-1}RS = \left[ \begin{matrix} 0.6 & -0.8\\ -0.8 & -0.6 \end{matrix} \right][/imath].)

The working is very lengthy, so I won't post it yet. If it is required, I will, but I don't think it will be. I just want to compare answers with someone else.

I got as my answers:
Line of invariant points is [imath]y = -0.5x[/imath]
Invariant line is [imath]y = 2x[/imath]

But the answer in the book simply states 'None'. Have I made a mistake somewhere in my working? (If so, I'll type all my working up or post a picture.) Or is the answer in the book wrong?

Huge thanks in advance.

 

[Dr.Peterson]

I got as my answers:
Line of invariant points is [imath]y = -0.5x[/imath]
Invariant line is [imath]y = 2x[/imath]

But the answer in the book simply states 'None'.

I haven't done any of the work, but just thinking about your answer in terms of the description of R and S, it seems correct. Rotating, reflecting, and rotating back takes your invariant points back to themselves, and takes points on your invariant line to other points on your invariant line, just as it should. I think the book is likely wrong.

 

[Vantage]

I haven't done any of the work, but just thinking about your answer in terms of the description of R and S, it seems correct. Rotating, reflecting, and rotating back takes your invariant points back to themselves, and takes points on your invariant line to other points on your invariant line, just as it should. I think the book is likely wrong.

Thank you. I'm pretty sure there are a few more errors in these answers, especially since the answer I got does come up, just not for the question I was doing. Maybe there was a mix up when writing up the answers? Who knows.

Regardless, thanks again for putting my mind at rest.