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Question

Prove that the matrix BAB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

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Solution

If A be symmetric i.e., A=A then
(BAB)=[B(AB)]=(AB)(B)
=(BA)B=BAB=BAB
Hence BAB is symmetric
If A is skew symmetric i.e., A=A then
(BAB)=[B(AB)]=(AB)(B)
=(BA)B=B(A)B=(BAB)
BAB is skew symmetric

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