Question

If A and B are symmetric matrices of same order, prove that AB + BA is a symmetric matrix

Answer 1

Given A and B are symmetric matrices

⇒ – A^T = A and B^T = B

To prove AB + BA is a symmetric matrix.

Proof: Now (AB + BA)^T = (AB)^T + (BA)^T 

= B^TA^T + A^TB^T

= BA + AB

= AB + BA

i.e. (AB + BA)^T = AB + BA

⇒ (AB + BA) is a symmetric matrix