So the question asks: Let $A$ and $B$ be n×nn×n orthogonal matrices, with $n≥2$. Which of the following matrices must be orthogonal?
A. $A^TB$
B. The matrix C obtained by multiplying the second column of $A$ by 3.
C. The matrix C obtained by switching the first two columns of $A$.
D. The matrix C obtained by adding the first column of $A$ to the second column of $A$.
E. $(AB)^{-1}$
F. $A+B$
So so far I got:
Since the product $AB$ of two orthogonal matrices is orthogonal, and the inverse of an orthogonal matrix is orthogonal. So E is right.
F does not have to be orthogonal. For example $I_2+I_2=2I_2$ which is not orthogonal.
Also, since $A^T=A^{-1}$ is an orthogonal matrix, then A is true.
But I have no idea anout B,C,D. I think C should be right because it does not change the number of the matrix. But is this true? And are B and D right and why?