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The way I have approached it so far has been a brute force method. I'm not really sure if this will be true or false, and I have a feeling it is false, but I can't construct a good counter-example. So, I have been trying to prove it is true, which is becoming tedious and lengthy as I go through the inner products.

I started on another more algebraic approach too. I know that [tex]Q^T Q = I[/tex], so if QQ = I, then [tex] Q = Q^{-1} = Q^T[/tex]. Also det(Q) = -1, since it is improper. From here I am not quite sure either.