<s>[INST] Relationships between $\det(A+B)$ and $A+B$ When computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$.
However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?
 [/INST] I can provide the following insight:
for every $A,B \in F^{n\times n}$
if we define $W=\left\{ M\in \left( \mathbb{Z} _{2}\right) ^{n\times n}| M \ is \ diagonal \right\} $ then the formal way of describing it is:
$$ det\left( A+B\right) =\sum _{M\in W}det\left( A\cdot M+B\cdot \left( I-M\right) \right) $$
in words, the sum of determinants above is the sum of all the determinants of the matrices that result from either removing the column from $B$ or from $A$ for each column in $A+B$ , the following is an example where $n = 2$ to better understand the idea:
$$
\left| A+B\right| =\begin{vmatrix}
a_{11}+b_{11} & a_{12}+b_{12} \\
a_{21}+b_{21} & a_{22}+b_{22}
\end{vmatrix}=\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{vmatrix}+\begin{vmatrix}
b_{11} & a_{12} \\
b_{21} & a_{22}
\end{vmatrix}+\begin{vmatrix}
a_{11} & b_{12} \\
a_{21} & b_{22}
\end{vmatrix}+\begin{vmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{vmatrix}
$$
we can notice that:
$$
\left| A+B\right| =det(A)+\begin{vmatrix}
\vert & \vert \\
B_{1} & A_{2}  \\
\vert & \vert 
\end{vmatrix}+\begin{vmatrix}
\vert & \vert \\
A_{1} & B_{2} \\
\vert & \vert
\end{vmatrix}+det(B)
$$
so in a way there is a relation between $det(A+B)$ and $det(A)+det(B)$ .
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