Question

If $A, B$ are symmetric matrices of the same order, then $AB - BA$ is a:

• A. Skew-symmetric matrix
• B. Symmetric matrix
• C. Zero matrix
• D. Identity matrix

Hint:

The commutator \(AB - BA\) of symmetric matrices is always skew-symmetric.

Answer 1

The Correct Option is A

Given \(A, B\) are symmetric matrices: \[ A^T = A, \quad B^T = B \] Consider: \[ (AB - BA)^T = B^T A^T - A^T B^T = BA - AB = -(AB - BA) \] Therefore: \[ (AB - BA)^T = -(AB - BA) \] This means \(AB - BA\) is skew-symmetric.