Question

Show that the matrix \(B'AB\) is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Answer 1

We suppose that A is a symmetric matrix, then A'=A… (1) 

Consider 
(B'AB)'={B'(AB)}' 
=(AB)'(B')'        [(AB)' =B'A'] 
=B'A'(B)            [(B')'=B] 
=B'(A'B) 
=B'(AB) 
therefore (B'AB)'=B'AB 

Thus, if A is a symmetric matrix, then 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')' 
=(B'A')B=B'(-A)B =-B'AB 
therefore (B'AB)'=-B'AB 

Thus, if A is a skew-symmetric matrix, then B'AB is a skew-symmetric matrix. Hence, if A is a symmetric or skew-symmetric matrix, then B'AB is a symmetric or skew symmetric matrix accordingly.