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Question
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Solution
(i) Let A be a symmetric matrix.
Then A’ = A
∴ (B’ AB) = (B’ (AB)) = (AB)'(B’)’
= (B’A’)B
=B’ AB [∵ (AB)’ = B’A’ and A’ = A]
⇒ B’ AB is a symmetric matrix.
(ii) Let A be a skew-symmetric matrix.
∴ A’ = -A
Now, (B'(AB))’ = (AB)’ (B’)’ = (B’A’)B
= B'(-A)B = -B’ AB [∵ A’ = -A]
= -(B’ AB)
Hence, B’ AB is a skew symmetric matrix.
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