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Question

Prove that the matrix $$B'AB$$ is symmetric or skew symmetric according as $$A$$ is symmetric or skew symmetric.


Solution

If $$A$$ be symmetric i.e., $$A'=A$$ then
$$(B'AB)'=[B'(AB)]'=(AB)'(B')'$$
$$=(B'A')B=B'A'B=B'AB$$
Hence $$B'AB$$ is symmetric
If $$A$$ is skew symmetric i.e., $$A'=-A$$ then
$$(B'AB)'=[B'(AB)]'=(AB)'(B')'$$
$$=(B'A')B=B'(-A)B=-(B'AB)$$
$$\therefore$$ $$B'AB$$ is skew symmetric

Mathematics

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