If A and B are symmetric matrices of same order, then which of the following are correct?
(A) \(AB - BA\) is a skew-symmetric matrix.
(B) \(AB + BA\) is a skew-symmetric matrix.
(C) \(AB^T - BA^T\) is a skew-symmetric matrix.
(D) \(AB + BA\) is a symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
Since A and B are symmetric, \(A^T = A\) and \(B^T = B\). By the reversal law:
\( \displaystyle (AB)^T = B^TA^T = BA \)
\( \displaystyle (BA)^T = AB \)
Then
\( \displaystyle (AB - BA)^T = BA - AB = -(AB - BA) \)
\( \displaystyle (AB + BA)^T = BA + AB = AB + BA \)
So \(AB - BA\) is skew-symmetric and \(AB + BA\) is symmetric.
Checking the statements
(A) \(AB - BA\) is a skew-symmetric matrix. — Correct. \((AB - BA)^T = -(AB - BA)\).
(B) \(AB + BA\) is a skew-symmetric matrix. — Incorrect. \((AB + BA)^T = AB + BA\), so it is symmetric (skew-symmetric only if it is the zero matrix).
(C) \(AB^T - BA^T\) is a skew-symmetric matrix. — Correct. Since \(B^T = B\) and \(A^T = A\), it is \(AB - BA\), which is skew-symmetric.
(D) \(AB + BA\) is a symmetric matrix. — Correct. \((AB + BA)^T = AB + BA\).