Which of the following statements is (are) true?

(A) $B^TAB$ is a skew-symmetric matrix if A is a symmetric matrix
(B) $B^TAB$ is a symmetric matrix if A is a symmetric matrix
(C) $B^TAB$ is a symmetric matrix if A is a skew-symmetric matrix
(D) $B^TAB$ is a skew-symmetric matrix if B is a skew-symmetric matrix
(E) $B^TAB$ is a symmetric matrix if B is a symmetric matrix

Choose the correct answer from the options given below:

(B) only

The correct answer is Option (3) → (B) only

(A) $B^T A B$ is skew-symmetric if $A$ is symmetric

Transpose gives $(B^T A B)^T = B^T A^T B = B^T A B$, hence the matrix is symmetric, not skew-symmetric.

False

(B) $B^T A B$ is a symmetric matrix if $A$ is a symmetric matrix

$(B^T A B)^T = B^T A^T B = B^T A B$ since $A^T=A$.

True

(C) $B^T A B$ is a symmetric matrix if $A$ is a skew-symmetric matrix

$(B^T A B)^T = B^T A^T B = -B^T A B$ since $A^T=-A$, hence it is skew-symmetric, not symmetric.

False

(D) $B^T A B$ is a skew-symmetric matrix if $B$ is a skew-symmetric matrix

Skew-symmetry of $B$ alone does not ensure skew-symmetry of $B^T A B$.

False

(E) $B^T A B$ is a symmetric matrix if $B$ is a symmetric matrix

No condition on $A$ is given, so symmetry of $B$ alone is not sufficient.

False

The correct statement is (B).