দেখুওৱা যে A B = A B যদি আৰু যদিহে A = B
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Answer:
Given:A andB are symmetric matrix
A
T
=A,B
T
=B
We need to show AB is symmetric if and only if A and B commute.
and if A and B commute (AB=BA), then AB is symmetric.
Part:I
If AB is symmetric then A and B commute.
Given AB is symmetric
⇒(AB)
T
=AB
⇒B
T
A
T
=AB using the property (AB)
T
=(BA)
T
⇒BA=AB (given A
T
=A and B
T
=B)
Hence A and B commute.
Hence proved.
Part:IIIf A and B commute, then AB is symmetric.
Given A and B commute.
i.e., AB=BA
We need to show AB is symmetric.
We need to show (AB)
T
=B
T
A
T
as
(AB)
T
=B
T
A
T
=BA as A=A
T
and B=B
T
given
=AB assumed that AB=BA
So, (AB)
T
=AB
Hence, AB is symmetric.
Hence proved.
Hence,AB is symmetric if and only if A and B commute, that is AB=BA
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