Show that the matrix B' AB is symmetric or skew-symmetric according to A which is symmetric or skew -symmetric.
We suppose that A is a symmetric matrix, then A'=A
Consider (B' AB)'=(B'(AB))'=(AB)'(B')' [∵(AB)′=B′A′]
=B′A′(B)[∵(B′)′=B]
=B′(A′B)=B′(AB)[∵A′=A]
∴(B′AB)′=B′A
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)′=B′A′and(A′)′=A]
=(B'A')B=B'(-A)B=-B'AB [∵A′=−A]
∴(B′AB)′AB which shows that B' AB is a skew -symmetric matrix.