Determinant of (A²+B²).

Question

Let $A$ and $B$ be distinct matrices. Knowing that $A^3=B^3$ and $A^2B=B^2A$, what's the value of $\det(A^2+B^2)$?

From the comment.
I end up finding that $\det(A^2+B^2)=\det(A)\det(A+B)$. I dont know how to proceed.

Answer 1

Honestly I didn't understand the hints that were in the comments. This is how one can reason. Since. $$ (A^2+B^2)A=A^3+B^2A=B^3+A^2B=(B^2+A^2)B, $$ then $(A^2+B^2)A=(A^2+B^2)B$. If $\det(A^2+B^2)\neq0$, then the matrix $A^2+B^2$ is invertible and hence $A=B$. Since by the condition $A\neq B$, we have $\det(A^2+B^2)=0$.