This is an exercise in linear algebra.
Suppose $A,B$ are $n\times n$ unitary matrices. Show that $|\det(A+B)|\leq2^n$.
When $n=1$, this is trivially true since by the definition of unitary matrices, we must have $|A|=|B|=1$ and we can check case by case that $|\det(A+B)|\leq2$.
But I am not able to proceed with the general case when $n>1$. All I know is that $\det(A)=\det(B)=1$ when $A$ and $B$ are both unitary. But since they are both matrices, I don't know how to relate this to the quantity $|\det(A+B)|$.
Can anyone help?