Question

Let A and B be two square matrices of the same dimension and let [A, B] = AB - BA. Then for three 2 × 2 matrices A, B, C, [[A, B], C] + [[B, C], A] + [[C, A], B] is equal to

(A) 1

(B) 0

(C) ABC - CBA

(D) None of these  

Answer 1

Answer is (B) 0

[[A, B], C] + [[B,C], A] + [[C,A], B]

= [AB - BA, C] + [BC - CB, A] + [CA - AC, B]

= (AB - BA)C - C(AB - BA) + (BC - CB)A - A(BC - CB) + (CA - AC)B