TITLE: Linear algebra and matrix.
QUESTION [5 upvotes]: prove or disprove : If A and B are 2 by 2 orthogonal matrices over R then A+B cannot be orthogonal.
OR 
If S,T:R^2--->R^2 are orthogonal transformation then S+T is not an orthogonal transformation.

REPLY [2 votes]: Counter-example is given. To show that the sum of orthogonal matricies can be orthogonal take 
$\begin{pmatrix}\frac{1}{2}&\frac{\sqrt3}{2}\\ -\frac{\sqrt3}{2}&\frac{1}{2}\end{pmatrix}+\begin{pmatrix}\frac{1}{2}&-\frac{\sqrt3}{2}\\ \frac{\sqrt3}{2}&\frac{1}{2}\end{pmatrix}$