Question

Let A and B both be $ 3\times 3 $ matrices. Then, $ {{(AB)}^{T}}=BA, $ if

• A. A is skew-symmetric and B is symmetric
• B. B is skew-symmetric and A is symmetric
• C. A and B are skew-symmetric
• D. None of the above

Answer 1

Answer: (C)

Let A and B are skew-symmetric.
$ \Rightarrow $ $ {{A}^{T}}=-A $ and $ {{B}^{T}}=-B $ ..(i)
Given, $ {{(AB)}^{T}}={{B}^{T}}{{A}^{T}} $
$ =(-B)(-A) $ [from Eq. (i)] $ =BA $