Upper bound for the determinant of matrix sum

Question

I wonder is there any way to express the "upper bound" (not lower bound) of $\det(A+B)$ in terms of $\det(A)$ and $\det(B)$ ? Thank you!

Answer 1

There is no upper bound for the determinant of $A+B$ in terms of the determinants of $A$ and $B$.

Let $A=\begin{pmatrix} k &k\\k&k\end{pmatrix}$ and $B=\begin{pmatrix} k &-k\\-k&k\end{pmatrix}$.