Question

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.

Answer 1

Hey there!

Here's the answer you are looking for!

Since AB is symmetric, we have
AB = $(AB)^T$

But $(AB)^T$ = $B^T$$A^T$

Hence, AB = $B^T$ $A^T$ ------(1)

It is also given that A and B are symmetric.

Hence by --(1) becomes,
AB = BA

Thus, AB is symmetric if and only if AB = BA.

HOPE IT HELPS ^_^

Answer 2

Hey there!

Here's the answer you are looking for!

Since AB is symmetric, we have

AB = (AB)^T(AB)

T

But (AB)^T(AB)

T

= B^TB

T

A^TA

T

Hence, AB = B^TB

T

A^TA

T

------(1)

It is also given that A and B are symmetric.

Hence by --(1) becomes,

AB = BA

Thus, AB is symmetric if and only if AB = BA.

HOPE IT HELPS ^_^