Determinants Matrices Question 150
Question: Which of the following is/are correct-
Options:
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
Show Answer
Answer:
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.
BETA
