I am interested in the follow expression
$$ \det (\exp M+ \exp N) $$
where $M$ and $N$ are $2\times 2$ real matrices (but also in the general case where the matrices are $n\times n$).
which represents a interference pattern that may have an application in Physics. I am trying to visualize the expression so that I can identify it with something familiar in nature.
The matrices are written as follows:
$$ M=\pmatrix{a+x & -b+y\\b+y & a-x}\\ N=\pmatrix{c+u & -z+v\\z+v & c-u} $$
Then I can write the matrices as $M=aI + V$ and $N=cI+W$.
In this case, $$ \begin{array} \det(\exp aI\exp V + \exp cI \exp W)\\ =(\exp aI\exp V + \exp cI \exp W)(\exp aI\exp -V + \exp cI \exp -W)\\ =(\exp aI)^2+(\exp cI)^2+2\exp aI\exp cI \exp W\exp -V \end{array} $$
How can I better visualize this result, considering that my hotel room is currently limited to three spacial dimensions.
edit:
actually the last term should be:
$$ \begin{array} =(\exp aI)^2+(\exp cI)^2+2\exp aI\exp cI (\exp W\exp -V + \exp V \exp -W)\\ =(\exp aI)^2+(\exp cI)^2+2\exp aI\exp cI (\exp -(V-W) + \exp (V -W)) \end{array} $$