Science:Math Exam Resources/Courses/MATH307/April 2005/Question 03 (a)
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Question 03 (a) 

Let P be a real symmetric n x n matrix such that P^{2} = P. Also let R = I_{n}  2P be the matrix that reflects a vector across a plane. (a) Show that R is an orthogonal matrix and that R^{2} = I_{n}. 
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Hint 

R is an orthogonal matrix if R^{T}R = RR^{T} = I. Calculate these matrix products and use that P is real symmetric, i.e. P^{T} = P. 
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Solution 

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Now since P is real symmetrical Also the transpose of the identity matrix, I is itself So expanding
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