Linear algebra and matrix.

Question

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.

Answer 1

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

Answer 2

$$ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = ? $$