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Question

Show that the matrix B' AB is symmetric or skew-symmetric according to A which is symmetric or skew -symmetric.

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Solution

We suppose that A is a symmetric matrix, then A'=A
Consider (B' AB)'=(B'(AB))'=(AB)'(B')' [(AB)=BA]
=BA(B)[(B)=B]
=B(AB)=B(AB)[A=A]
(BAB)=BA
Which show that B' AB is a symmetric matrix.
Now, we suppose that A is a skew-symmetric matrix.
Then, A'=-A
Consider (B'AB)'=(B'(AB))'=(AB)'(B') [(AB)=BAand(A)=A]
=(B'A')B=B'(-A)B=-B'AB [A=A]
(BAB)AB which shows that B' AB is a skew -symmetric matrix.


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