Question

Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.

Options

• skewsymmetric

• symmetric

• neither symmetric nor skewsymmetric

• I or – I, where I is an identity matrix

Answer 1

Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is symmetric.

Explanation:

Let A be symmetric matrix and B be skew-symmetric matrix.

∴ A^T = A and B^T = –B

Consider

(AB – BA)^T = (AB)^T – (BA)^T

= B^TA^T – A^TB^T

= (–B) (A) – (A) (–B)

= –BA + AB

= AB – BA

This shows AB – BA is symmetric matrix.