The commutator of two matrices A, B, is defined to be the matrix
C = [A.B] = AB - BA. (1.12)
(a) Explain why [A.B] is defined if and only if A and B are square matrices of the same size.
(b) Show that A and B commute under matrix multiplication if and only if [Aˆ™ B] = O.
(c) Compute the commutator of the following matrices:
(i)

(ii)

(iii)

(d) Prove that the commutator is
(i) Bilinear.

for any scalars c. d;
(ii) Skew-symmetric; [A, B] = - [B.A];
(iii) Satisfies the the Jacobi identity;

for any square matrices A.B.C of the same size.

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