If A and B are symmetric matrices of the same order then which of the following statements are correct?
(A) AB is symmetric matrix
(B) AB − BA is skew symmetric matrix
(C) \( B^TAB \) is symmetric matrix
(D) \( A-A^T \) is skew symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (D) only
Since A and B are symmetric, \( A^T=A \) and \( B^T=B \). A matrix M is symmetric if \( M^T=M \) and skew symmetric if \( M^T=-M \). By the reversal law \( (PQ)^T=Q^TP^T \):
\( \displaystyle (AB)^T=B^TA^T \)
\( \displaystyle =BA \)
\( \displaystyle (AB-BA)^T=BA-AB \)
\( \displaystyle =-(AB-BA) \)
\( \displaystyle (B^TAB)^T=B^TA^T(B^T)^T \)
\( \displaystyle =B^TAB \)
\( \displaystyle (A-A^T)^T=A^T-A \)
\( \displaystyle =-(A-A^T) \)
So \( (AB)^T=BA \), which equals AB only when A and B commute; AB − BA and \( A-A^T \) change sign on transposing; and \( B^TAB \) is unchanged.
Checking the statements
(A) AB is symmetric matrix — Incorrect. \( (AB)^T=BA \), which need not equal AB: for \( A=\begin{bmatrix}1 & 0\\0 & 2\end{bmatrix} \), \( B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} \), \( AB=\begin{bmatrix}0 & 1\\2 & 0\end{bmatrix} \), which is not symmetric.
(B) AB − BA is skew symmetric matrix — Correct. Its transpose is BA − AB = −(AB − BA).
(C) \( B^TAB \) is symmetric matrix — Correct. Its transpose is \( B^TAB \) itself.
(D) \( A-A^T \) is skew symmetric matrix — Correct. Its transpose is \( A^T-A=-(A-A^T) \); here it is the zero matrix, which is skew symmetric.