$A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$ I found the following problem interesting but do not know how to tackle it. 

If $A$ and $B$ are different  matrices satisfying $A^3=B^3$ and $A^2B=B^2A$.Then find $\det (A^2+B^2)=?.$ 

Can someone point me in the right direction?

Since $(A^2+B^2)(A-B)=A^3-A^2B+B^2A-B^3=0_n.$ Since $A \neq B,$ this shows that  $A^2+B^2$ has a zero divisor.Hence it 
is not invertible. So its determinant is $0.$
