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Let A and B be two nonsingular square matrices, `A^(T)` and `B^(T)` are the tranpose matrices of A and B, respectively, then which of the following are coorect ?
A. `B^(T)AB` is symmetric matrix if A is symmetric
B. `B^(T)AB` is symmetric matrix if B is symmetric
C. `B^(T)AB` is skew-symmetric matrix for every matrix A
D. `B^(T)AB` is skew-symmetric matrix if A is skew-symmetric

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1 Answers

Correct Answer - A::D
`(B^(T)AB)^(T)=B^(T)A^(T) (B^(T))^(T)=B^(T)A^(T)B=B^(T)AB` if `A` is symmetric.
Therefore, `B^(T)AB` is symmetric if A is symmetric.
Also, `(B^(T)AB)^(T)=B^(T)A^(T)B=B^(T) (-A) B=-(B^(T)A^(T)B)`
Therefore, `B^(T) AB` if A is skew-symmetric if A is skew-symmetric.

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