<s>[INST] Find $\det(A^3B^2+A^2B^3)$ for $A,B$ s.t. $AB = BA$ given $\det A,\det B$. If two matrices $A$ and $B$ are such that $AB = BA$ with $\det A = 1$ and $\det B = 0$, then what is $\det(A^3B^2 + A^2B^3)$ ? 
 [/INST] \begin{align*}
\text{det}(A^3B^2+A^2B^3) & = \text{det}A^2 \text{det} B^2\text{det}(A+B)\\
& = (\text{det}A)^2 (\text{det} B)^2\text{det}(A+B)\\
&=0.
\end{align*}
</s>