By using matrix method, show that a reflection in the line $y=x$ followed by a reflection in $Y$ axis is equivalent to the rotation through $90$ about the origin.
My attempt
Am I going right? Or, is there any other alternative?
Question on proving the single transformation equivalent to combined transformation using matrix
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$\begingroup$
matrices
transformation
rotations
reflection
1 Answers
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In an answer to a question like this, I would expect to see
- The matrix for the reflection across the line $y=x$ written out in explicit numbers.
- The matrix for the reflection across the $y$ axis written out in explicit numbers.
- Some matrix arithmetic using the previous two matrices, resulting in the rotation matrix written out in explicit numbers.
- All of the above matrices would be clearly identified by naming the transformations they performed.
What you have appears to be a good derivation of the first matrix mentioned above. I would only caution that while the series of four equations you wrote in order to derive the matrix may help you conceptually with this exercise, I'm not sure it will help someone check your work (or grade your paper). I don't think I would really need to see the general matrix form $\begin{pmatrix} a&b\\c&d \end{pmatrix}$ written out four times; what I would want to see is a demonstration that you made sure that you had the right matrix, that is, that $$\begin{pmatrix} 0&1\\1&0 \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} y\\x \end{pmatrix}.$$ You have demonstrated that fact in your working, but it's a little bit obscured by being written among so many equations. What we really need to know is just that the right-hand sides of the first and third equations are equal.
In short, your work looks correct to me on careful examination, but it's harder to check than it needs to be. It may be better to do this work on scrap paper and then transfer the necessary pieces of information (including enough to verify of the result) to your homework or exam paper.