Let P and Q be 3x3 matrices with P≠Q If P3=Q3  and P2Q=Q2P then determinant of P2+Q2 is equal to

P3=Q3 and P2Q=Q2P givesP3−P2Q=Q3−Q2P⇒ P2(P−Q)=−Q2(P−Q)⇒P2+Q2(P−Q)=0If det P2+Q2≠0 then P2+Q2 is invertible and hence P=Q. Therefore, det⁡P2+Q2=0.