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দেখুওৱা যে A B = A B যদি আৰু যদিহে A = B |
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Answer» Given:A ANDB are symmetric matrixAT=A,BT=BWe 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:IIF AB is symmetric then A and B commute.Given AB is symmetric⇒(AB)T=AB⇒BTAT=AB USING the property (AB)T=(BA)T⇒BA=AB (given AT=A and BT=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=BAWe need to show AB is symmetric.We need to show (AB)T=BTATas(AB)T=BTAT=BA as A=ATand B=BTgiven=AB assumed that AB=BASo, (AB)T=ABHence, 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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