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Kishore Kumar

Feb 2, 2014

If A and B are symmetric matrices of the same order

If A and B are symmetric matrices of the same order, then show that AB is symmetric if and only if A and B commute, that is AB = BA

Answers(1)

Answer

Kishore Kumar

Since AB is symmetric, we have AB = (AB)T
But (AB)T = BT AT
Hence AB = BT A…(1)
It is also given that A and B are symmetric
A= A and BT = B
Hence (1) becomes, AB = BA
Thus AB is symmetric if and only if AB = BA.
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