Are row (or column) sums of orthogonal matrix (excluding scalar multiples of identity matrix) always different?
Suppose $Q$ is an $p\times p$ orthogonal matrix that is not a scalar multiple of identity matrix and $\mathbf{1}_p=(1,1,\ldots,1)^T$ which is a column vector of ones. Then is $Q\cdot\mathbf{1}_p\neq k\mathbf{1}_p$ always true for arbitary $k$?
If not, is there any counterexample?
I have no clue how to solve this other than the intuition that rotations and reflections are represented by orthogonal matrix so that the statement is likely to be true.
Edit: $Q$ should not be identity matrix $I$.
Edit2: $Q$ should not be scalar multiples of identity matrix $I$.