A) B' AB is symmetric if A is symmetric
B) B' AB is skew-symmetric if A is symmetric
C) B' AB is symmetric if A is skew-symmetric
D) None of these
Correct Answer: A
Solution :
[a] Let A be a symmetric matrix. Then, \[A'=A\] Now, \[(B'AB)'=B'A'(B')'.\] \[[\because (AB)'=B'A']\] \[=B'A'B[\because (B)'=B]\] \[=B'AB[\because A'=A]\] \[\Rightarrow B'AB\] is a symmetric matrix. Now, let A be a skew-symmetric matrix. Then, \[A'=-A\] \[\therefore (B'AB)'=B'A'(B')'[\because (AB)'=B'A']\] \[=B'A'B[\because (B')'=B]\] \[=B'(-A)B[\because A'=-A]\] \[=-B'AB\therefore B'AB\] is a skew-symmetric matrix.
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