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
By the reversal law for transposes, \((XYZ)^T = Z^TY^TX^T\), and \((B^T)^T = B\):
\((B^TAB)^T = B^TA^TB\)
So the nature of \(B^TAB\) depends on \(A^T\), not on B.
Checking the statements
(A) \(B^TAB\) is a skew-symmetric matrix if A is a symmetric matrix — Incorrect. With \(A^T = A\), \((B^TAB)^T = B^TAB\), so it is symmetric, not skew-symmetric (unless it is the zero matrix).
(B) \(B^TAB\) is a symmetric matrix if A is a symmetric matrix — Correct. With \(A^T = A\), \((B^TAB)^T = B^TAB\).
(C) \(B^TAB\) is a symmetric matrix if A is a skew-symmetric matrix — Incorrect. With \(A^T = -A\), \((B^TAB)^T = -B^TAB\), so it is skew-symmetric.
(D) \(B^TAB\) is a skew-symmetric matrix if B is a skew-symmetric matrix — Incorrect. Nothing is known about A. For example, with A = I and
\( B = \begin{bmatrix}0&1\\-1&0\end{bmatrix},\quad B^TIB = B^TB = I \)
which is symmetric, not skew-symmetric.
(E) \(B^TAB\) is a symmetric matrix if B is a symmetric matrix — Incorrect. Again A is arbitrary. With B = I, \(B^TAB = A\), which need not be symmetric.