Matrices & Determinants

$Problem$ #1 If $A$ and $B$ are different matrices satisfying $A^3 = B^3$ and $A^2B = B^2A$, find $det(A^2+B^2)$ Since $A$ and $B$ are different matrices $A-B \neq O$, Now $(A^2+B^2)(A-B) = A^3-A^2B+B^2A-B^3$ =$O$ since $A^3 = B^3$ and $A^2B = B^2A$ This shows that $(A^2+B^2)$ has a zero divisor, so it is not invertible hence $det(A^2+B^2) = 0$