Fill in the blanks in each of the
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.

Question

Fill in the blanks in each of the If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.

Philoid · Accepted Answer

This is only possible if A and B commute.

Proof:

Given A and B are symmetric matrices,

⇒ A’=A ..(1)

⇒ B’=B ..(2)

Let AB is a Symmetric matrix:-

⇒ (AB)’=AB

Using Property (AB)’=B’A’

⇒ B’A’=AB

⇒ Now using (1) and (2)

⇒ BA=AB

Hence A and B matrix commute.

Fill in the blanks in each of theIf A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.