Determinant algebra If $A$ and $B$ are $4 \times 4$ matrices with $\det(A) = −2$, $\det(B) = 3$, what is $\det(A+B)$?
At first I approached the problem that $\det(A+B) = \det(A) + \det(B)$ but this general rule would not hold true, so I do not know how to approach the problem from here.

The number $\det(A+B) $ is not determined by the determinants of $A$ and $B$. For instance if
$$
A=\begin{bmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-2\end{bmatrix}, \ \ 
B=\begin{bmatrix} -1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&3\end{bmatrix}
$$
then $A$ and $B$ are as required, with $\det A+B=0$. While if$$
A=\begin{bmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-2\end{bmatrix}, \ \ 
B=\begin{bmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&3\end{bmatrix}
$$
then $\det A+B=8$.
