Science:Math Exam Resources/Courses/MATH221/April 2013/Question 07 (c)

MATH221 April 2013

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[hide] Question 07 (c)
Let $T:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the linear transformation that reflects points through the line 3x = 4y. c) Compute A1995.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint
Consider what happens to a vector after successive reflections - it may help to write down a sample vector and it's first 3 or 4 reflections. Do you notice a pattern?

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[show] Solution 1
From the description of A, applying the matrix 1995 to $R^{2}$ is simply applying the same reflection 1995 times, but as reflections have order 2, i.e. applying it twice does nothing, and 1995 is an odd number, we see that $A^{1995}=A$.

[show] Solution 2

Alternatively, we can also show this by straightforward computation, using the breakdown in part b): ${\begin{aligned}A^{1995}&=\left(B\left[{\begin{array}{cc}1&0\\0&-1\end{array}}\right]B^{-1}\right)^{1995}\\&=B\left[{\begin{array}{cc}1&0\\0&-1\end{array}}\right]^{1995}B^{-1}\\&=B\left[{\begin{array}{cc}1^{1995}&0\\0&(-1)^{1995}\end{array}}\right]B^{-1}\\&=B\left[{\begin{array}{cc}1&0\\0&-1\end{array}}\right]B^{-1}\\&=A\end{aligned}}$

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