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:
Answer & explanation
Correct answer: option 3
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).