Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 06 (c)
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Question B 06 (c) 

The matrix A below represents a transformation that reflects vectors in in a plane that contains the origin. Note: you can use geometric insight to greatly simplify the calculations below. (c) Find for all integers . 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Imagine geometrically when you reflects a vector times with respect to some plane. 
Hint 2 

(For Solution 2,) diagonalize to get 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Geometrically, when you reflects a vector times with respect to some plane, it becomes itself for even and become its reflection for odd . Since the matrix is the operation reflecting a vector in some plane (obtained in part (b)), for any vector in , for even and for odd . Therefore, we have

Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. (Alternative solution) By part (a), we know that where and Then Thus 