Let `Aa n dB` be two nonsingular square matrices, `A^T a n dB^T` are the transpose matrices of `Aa n dB ,` respectively, then which of the following options are correct? (correct option may be more than one) `(a)`. `B^T A B` is symmetric matrix `(b)`. if `A` is symmetric `B^T A B` is symmetric matrix `(c)`. if `B` is symmetric `B^T A B` is skew-symmetric matrix for every matrix `A` `(d)`.`B^T A B` is skew-symmetric matrix if `A` is skew-symmetric

Options

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

Answer

A, D

Explanation

`(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.