Let $A, B$ be two orthogonal matrices over a field $F$ of characteristic $2$ such that $$\det (A) + \det (B) = 0.$$ Is $(A+B)$ necessarily a singular matrix?
I have proved the result to be true for real matrices anf the result also holds for complex matrices. I even proved the result over any field of characteristic $\neq 2.$ Can it hold for matrices over a field of characteristic $2$?
I am asking this question because at the fag end of the proof of this result for real matrices I got a relation $2 \det (A + B) = 0,$ since $2 \neq 0$ over $\Bbb R$ we have the required result. But for any field $F$ of characteristic $2$ we have $2 = 0$ and hence we can't say whether or not $\det (A+B) = 0$ so that $(A+B)$ is a singular matrix.
Any help or suggestion in this regard will be highly appreciated. Thanks in advance.
Here's a partial answer, namely specifically for $F=\Bbb F_2$: Here, $A^TA=I$ means that every column of $A$ has an odd number of $1$-entries. That is, with $v=\sum_ie_i$ as the all-1-vector, we have $v^TAe_i=1$ for all base vectors $e_i$. Same for $B$. But then $v^T(A+B)e_i=0$ for all base vectors, i.e., the image of $A+B$ is a proper subspace, meaning $A+B$ is singular.