<s>[INST] 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.
 [/INST] Let 
$$
A
=
\begin{pmatrix}
-2 & 0 & 0 & 0 \\
0  & 1 & 0 & 0 \\
0  & 0 & 1 & 0 \\
0  & 0 & 0 & 1 
\end{pmatrix}
\quad
B=
\begin{pmatrix}
3 & 0 & 0 & 0 \\
0  & 1 & 0 & 0 \\
0  & 0 & 1 & 0 \\
0  & 0 & 0 & 1 
\end{pmatrix}
\quad
C=
\begin{pmatrix}
-3  & 0 & 0 & 0 \\
0  & -1 & 0 & 0 \\
0  & 0 & 1 & 0 \\
0  & 0 & 0 & 1 
\end{pmatrix}
$$
Then $\det (A)=-1$ and $\det( B)=\det( C)=3$. However,
$$
\det(A+B)=8
$$
while
$$
\det(A+C)=0
$$
Your problem is not solvable!
</s>