Let A, B, C be n n real matrices and are pairwise commutative and ABC = O _ n and if = det (A^3+B^3+C^3 ) det (A+B+C) then

Step-by-step solution

aligned & A^3+B^3+C^3=A^3+B^3+C^3-3 A B C=(A+B+C) (A^2+B^2+C^2-A B-B C-C A ) & =(A+B+C) (A+ B+ ^2 C ) (A+ ^2 B+ C ) = cube root of unity aligned =(A+B+C) (A+ B+ ^2 C ) ( A+ B+ ^2 C ) aligned & =(A+B+C) (A+ B+ ^2 C ) (A+ B+ ^2 C ) & then det (A^3+B^3+C^3 ) det (A+B+C)= det (A+B+C)^2 det (A+ B+ ^2 C ) det (A+ B+ ^2 C ) & =( det (A+B+C))^2 | det (A+ B+ ^2 C ) |^2 0 aligned