Answer
The statement $det(A+B)=detA+detB$ is sometimes true, as shown below.
Work Step by Step
$A=\begin{bmatrix}
1&2&3&4&5\\
2&3&4&5&6\\
3&4&5&6&7\\
4&5&6&7&8\\
5&6&7&8&9
\end{bmatrix}$
$detA=0$
$B=\begin{bmatrix}
5&4&3&2&1\\
6&5&4&3&2\\
7&6&5&4&3\\
8&7&6&5&4\\
9&8&7&6&5
\end{bmatrix}$
$detB=0$
$detA+detB=0$
$A+B=\begin{bmatrix}
6&6&6&6&6\\
8&8&8&8&8\\
10&10&10&10&10\\
12&12&12&12&12\\
14&14&14&14&14
\end{bmatrix}$
$det(A+B)=0$
$A=\begin{bmatrix}
2&4&1&5\\
1&8&4&6\\
0&-4&5&6\\
4&0&-5&14\\
\end{bmatrix}$$detA=996$
$B=\begin{bmatrix}
4&3&8&-1\\
6&-4&7&13\\
0&-6&3&-5\\
1&-2&0&6
\end{bmatrix}$$detB=-148$
$detA+detB=848$
$A+B=\begin{bmatrix}
6&7&9&4\\
7&4&11&19\\
0&-10&8&1\\
5&-2&-5&20
\end{bmatrix}$$det(A+B)=12970$
$A=\begin{bmatrix}
5&-4&0\\
1&0&3\\
4&5&10\\
\end{bmatrix}$$detA=-83$
$B=\begin{bmatrix}
-7&4&-3\\
-9&8&7\\
-7&16&1\\
\end{bmatrix}$$detB=832$
$detA+detB=749$
$A+B=\begin{bmatrix}
-2&0&-3\\
-8&8&10\\
-3&21&11\\
\end{bmatrix}$$det(A+B)=676$
$A=\begin{bmatrix}
5&-4\\
1&4\\
\end{bmatrix}$$detA=24$
$B=\begin{bmatrix}
-7&4\\
-9&8\\
\end{bmatrix}$$detB=-20$
$detA+detB=4$
$A+B=\begin{bmatrix}
-2&0\\
-8&12\\
\end{bmatrix}$$det(A+B)=-24$
