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Question
Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix
Solution
Let A and B be two symmetric matrices
⇒ AT = A and BT = B ........(1)
Given that AB = BA (2)
To prove AB is symmetric:
Now (AB)T = BTAT = BA
(From(1)) But (AB)T = AB by ........(2)
⇒ AB is symmetric.
Conversely let AB be a symmetric matrix.
⇒ (AB)T = AB
i.e. BTAT = AB
i.e. BA = AB .......(From (1))
⇒ AB is symmetric
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