Question

Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew-symmetric, then the matrix AB - BA is

(A) skew-symmetric

(B) symmetric

(C) neither symmetric nor skew-symmetric

(D) I or - I, where I is an identity matrix

Answer 1

Answer is (B) symmetric

 We have

(AB - BA)′ = (AB)′ - (BA)′ = B′A′ - A′B′ = (-B)(A) - A(-B) = AB - BA 

Therefore, AB - BA is symmetric