If A and B are symmetric matrices of the same order, then which of the following are true?
(A) AB − BA is a skew symmetric matrix
(B) AB is a symmetric matrix
(C) AB is a scalar matrix
(D) AB + BA is a symmetric matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
Since A and B are symmetric, \(A^T = A\) and \(B^T = B\). By the reversal law of transposes:
\((AB)^T = B^T A^T = BA\)
\((BA)^T = A^T B^T = AB\)
Hence:
\((AB - BA)^T = BA - AB = -(AB - BA)\)
\((AB + BA)^T = BA + AB = AB + BA\)
AB itself need not be symmetric or scalar. For example, with
\( A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix},\quad B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \)
\( AB = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \)
So AB − BA is skew symmetric, AB + BA is symmetric, and AB is in general neither symmetric nor scalar.
Checking the statements
(A) AB − BA is a skew symmetric matrix — Correct. Its transpose is \(-(AB - BA)\).
(B) AB is a symmetric matrix — Incorrect. \((AB)^T = BA\), which equals AB only when A and B commute.
(C) AB is a scalar matrix — Incorrect. A scalar matrix has all off-diagonal entries 0, but for the A and B taken here, AB has the off-diagonal entry 1.
(D) AB + BA is a symmetric matrix — Correct. Its transpose is \(BA + AB = AB + BA\).