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.
"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.